How Big Is Infinity?(quantamagazine.org)
quantamagazine.org
How Big Is Infinity?
https://www.quantamagazine.org/how-big-is-infinity-20220927/
28 comments
Diagonalization might be a tricky topic for beginners, but that is also who we teach it to. It is usually in your introduction to mathematical proofs class, and even there it is taught in the first couple of weeks.
There is an additional problem with the argument as written, as there are in fact "relations" on the "decimal expansions" of real numbers. for instance .099... = .1000....
Notice that none of the places in this expansion agree with each other so this might indeed be the element you construct from the following list:
`.1000... .0100... .0010... ...`
You might say, ok well pick a number different from a_i and also different from 0 or 9, which does indeed get you a number not in your list, but it really begs the question of "why doesn't this work in binary?", and is also far less intuitive than factoring the problem into two steps of |N|~=|P(N)|=|R|. This approach is more general anyway.
There is an additional problem with the argument as written, as there are in fact "relations" on the "decimal expansions" of real numbers. for instance .099... = .1000....
Notice that none of the places in this expansion agree with each other so this might indeed be the element you construct from the following list:
`.1000... .0100... .0010... ...`
You might say, ok well pick a number different from a_i and also different from 0 or 9, which does indeed get you a number not in your list, but it really begs the question of "why doesn't this work in binary?", and is also far less intuitive than factoring the problem into two steps of |N|~=|P(N)|=|R|. This approach is more general anyway.
> for instance .099... = .1000
wait, so how does diagonalization account for this? The argument, as i recall, was that every real number in the "fake list" differs from the generated real number by at least 1 position in the decimal expansion.
But doesn't this argument depend on the assumption that every real number has a unique decimal expansion?
wait, so how does diagonalization account for this? The argument, as i recall, was that every real number in the "fake list" differs from the generated real number by at least 1 position in the decimal expansion.
But doesn't this argument depend on the assumption that every real number has a unique decimal expansion?
Well, the short answer is that there aren't that many related decimal expansions at most two for any expansion, and a countable number of pairs, but it's a problem with the argument.
I was equally annoyed by this part. You are very generous with your very close to making an incorrect statement. I would straight away say it’s both misleading and wrong.
The author specifically states, and I quote:
>no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between ... this doesn’t mean that the sets of real numbers and natural numbers have different sizes
What is misleading or wrong about that? What specific statement is incorrect?
>no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between ... this doesn’t mean that the sets of real numbers and natural numbers have different sizes
What is misleading or wrong about that? What specific statement is incorrect?
No matter how close any two distinct rational numbers are to each other, there will always be infinitely many rational numbers in between. But the cardinality of rational numbers and natural numbers is the same.
That's correct and it's precisely what the author states, and I quote:
>this doesn’t mean that the sets of real numbers and natural numbers have different sizes
>this doesn’t mean that the sets of real numbers and natural numbers have different sizes
They state that only as a disclaimer, after a bunch of other statements that hint at the opposite. Just adding a disclaimer to misleading writing merely makes it confusing and unclear for those who are paying attention, and still misleading for those who don't notice the disclaimer.
I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.
I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.
I would say your criticism is akin to the curse of knowledge [1], wherein you know all of the relevant background information and so you have difficulty putting yourself in the position of someone who does not.
The author is writing the article for someone who could very well be learning this for the first time and may think to themselves "Hmm... this is something unusual and counterintuitive.". Since you are not that reader the author's writing could come across as annoying, but it's not wrong or misleading, it's a way for the author to hint to the reader that they are empathizing with them and will address this unintuitive notion further (which the author does).
[1] https://en.wikipedia.org/wiki/Curse_of_knowledge
The author is writing the article for someone who could very well be learning this for the first time and may think to themselves "Hmm... this is something unusual and counterintuitive.". Since you are not that reader the author's writing could come across as annoying, but it's not wrong or misleading, it's a way for the author to hint to the reader that they are empathizing with them and will address this unintuitive notion further (which the author does).
[1] https://en.wikipedia.org/wiki/Curse_of_knowledge
That’s an interesting quote you got there. Let’s look at the surrounding sentences, shall we? Emphasis mine.
> But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers.
> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation.
So no, as rightfully pointed by other commenters, the article implies that density somewhat indicates that two cardinalities which is straight wrong as the rational are dense in the natural yet of the same cardinality. They are even dense in R which entirely defeat the point of the analogy. Density and cardinality are not related at all.
> But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers.
> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation.
So no, as rightfully pointed by other commenters, the article implies that density somewhat indicates that two cardinalities which is straight wrong as the rational are dense in the natural yet of the same cardinality. They are even dense in R which entirely defeat the point of the analogy. Density and cardinality are not related at all.
> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation.
This seems a really odd example to start with, without coming back to, because the same is true of the rationals and they do have the same cardinality as the naturals.
This seems a really odd example to start with, without coming back to, because the same is true of the rationals and they do have the same cardinality as the naturals.
The distinction between the size of {I} and {I}/{I} (and {I}^n) is not the size, but the orderability.
(Sorry for the bullshit syntax.)
You can map any rational onto an integer index; you cannot do it so that that r[i - 1] < r[i] < r[i + 1].
This has no impact on cardinality, but there's something here that doesn't sit quite right with me.
(Sorry for the bullshit syntax.)
You can map any rational onto an integer index; you cannot do it so that that r[i - 1] < r[i] < r[i + 1].
This has no impact on cardinality, but there's something here that doesn't sit quite right with me.
Note, you also can't do this type of monotonic mapping with the integers mapped to the natural numbers. So its not like the rationals are a dividing point here.
Isn't it really the same thing?
The closest mapping of the reals to a countable set is the set of rational numbers.
Or am I an idiot?
The closest mapping of the reals to a countable set is the set of rational numbers.
Or am I an idiot?
There's room for an infinity of closer mappings.
E.g ratios of integers and square roots.
E.g ratios of integers and square roots.
Oh damn.
That's some sneaky code injection.
Basically you can add a countable infinity of new countable infinite dimensions to any countably infinite set and get something that's still countably infinite.
That shit still feels wrong to me.
That's some sneaky code injection.
Basically you can add a countable infinity of new countable infinite dimensions to any countably infinite set and get something that's still countably infinite.
That shit still feels wrong to me.
What doesn't sit right?
Your orderability idea is about "monotonic" mappings.
Rational numbers are interesting even if they aren't ordered at all (but still can add and multiply, like other non-ordered objects like the integers mod N).
Your orderability idea is about "monotonic" mappings.
Rational numbers are interesting even if they aren't ordered at all (but still can add and multiply, like other non-ordered objects like the integers mod N).
I don't quite know.
It feels as if there's some deep structural aspects to the rationals that get casually tossed to the side in order to shoehorn them into integer indices.
It feels as if there's some deep structural aspects to the rationals that get casually tossed to the side in order to shoehorn them into integer indices.
I guess the point is that you can't map them in a way that preserves "structure" in some sense.
Yes, seems like a mistake to mention that without saying that the difference is that they have different order types (and that there is a sense of some order types being "much bigger" than others, even if they have the same cardinality)
I always disliked this question. It's explicitly an exercise in futility.
The real answer to the question is to point out that the question is broken. It's pitting the prescriptive against the descriptive, then acting surprised they aren't the same thing.
The question "how big" only works with the set of "quantifiable". It's just a type error. Yet it's unsatisfying to say infinity isn't quantifiable, because when we do, we aren't being descriptive. Infinity is by definition unquantifiable, which is a prescriptive statement. Prescriptive answers just aren't any fun. We aren't learning anything from them, because we knew before we asked.
When we talk about "bigger and smaller" infinity, we are just using infinity as an abstraction in the very same way we use variables. It's just as straightforward as going from "x+1>x" to "+1>". We all know that "plus one is more". The first statement is using nouns, and the second is using functions. It's just a type difference, nothing more.
The thing we are spending so much time blathering in awe about is just the relative difficulty in describing abstraction. Abstraction is amazing, impressive, useful, often surprising or elegant. It is not however, mythical.
There's this thing we do where what we are talking about doesn't have any substance. It's called nonsense. That's it. There is no "deeper meaning" behind the explicit absence of meaning. It can be entertaining to talk in circles, but we know they aren't getting us anywhere new.
The real answer to the question is to point out that the question is broken. It's pitting the prescriptive against the descriptive, then acting surprised they aren't the same thing.
The question "how big" only works with the set of "quantifiable". It's just a type error. Yet it's unsatisfying to say infinity isn't quantifiable, because when we do, we aren't being descriptive. Infinity is by definition unquantifiable, which is a prescriptive statement. Prescriptive answers just aren't any fun. We aren't learning anything from them, because we knew before we asked.
When we talk about "bigger and smaller" infinity, we are just using infinity as an abstraction in the very same way we use variables. It's just as straightforward as going from "x+1>x" to "+1>". We all know that "plus one is more". The first statement is using nouns, and the second is using functions. It's just a type difference, nothing more.
The thing we are spending so much time blathering in awe about is just the relative difficulty in describing abstraction. Abstraction is amazing, impressive, useful, often surprising or elegant. It is not however, mythical.
There's this thing we do where what we are talking about doesn't have any substance. It's called nonsense. That's it. There is no "deeper meaning" behind the explicit absence of meaning. It can be entertaining to talk in circles, but we know they aren't getting us anywhere new.
> Prescriptive answers just aren't any fun. We aren't learning anything from them, because we knew before we asked.
All of mathematics is arguably a tautology. I think what you are saying applies to everything in math all the way back to 1+1
All of mathematics is arguably a tautology. I think what you are saying applies to everything in math all the way back to 1+1
Every thing, yes, but the things aren't what make math interesting.
It's the way those things relate to each other that is so interesting. The patterns. The connections.
It's the same with this discussion about infinity. The real substance in most of this article isn't infinity: it's set theory. Infinity is just being used as the hook to grab your attention.
It's the way those things relate to each other that is so interesting. The patterns. The connections.
It's the same with this discussion about infinity. The real substance in most of this article isn't infinity: it's set theory. Infinity is just being used as the hook to grab your attention.
Math on it's own, with concepts like this, can get a bit abstract.
I like connecting the concept with something concrete. For example the infinity of time. I once watched a fascinating documentary on the end of the universe, that talked about what would happen if the universe were to keep expanding forever and ever. first all the stars burn out, then a bunch of blackholes form, thenthe blackholes dissipate, etc,etc. they mentioned that actual protons would eventually breakdown after many 10 to the 10s etc.
I like connecting the concept with something concrete. For example the infinity of time. I once watched a fascinating documentary on the end of the universe, that talked about what would happen if the universe were to keep expanding forever and ever. first all the stars burn out, then a bunch of blackholes form, thenthe blackholes dissipate, etc,etc. they mentioned that actual protons would eventually breakdown after many 10 to the 10s etc.
I feel like that is a different concept of infinity than the one the article is talking about. Thinking about time as it approaches infinity feels more like an asymptotic notion of infinity instead of a cardinality one, i.e. similar to the type of infinity we mean when we say quicksort is O(n log n).
That said, i dont know how to really make this any more concrete then what the article did. Perhaps an analogy of an infinitely precise ruler with every possible tick marked off (e.g. marking off 0.1 cm, 0.01, and so on for every real number), and how if in addition to labeling the ticks with lengths we also wanted to label them as first tick, second tick, third tick, it would be impossible. The basic gist of the proof from the article is given any proposal for a way to label the ticks on our ruler as first, second, third, ... covering all the ticks, there is a generic way to find a tick that is missed, so any such proposal must miss some ticks. So in a sense the real numbers are "bigger" than the integers, because if you try to match them up you will always have real numbers left over. Thus people say the real numbers are uncountable as there is no way to count them all.
This is in contrast to something like the integers where you could say 0 is the first one, 1 is second, -1 is third, 2 is fourth and so forth, eventually getting to all of them given infinite time with nothing left over.
That said, i dont know how to really make this any more concrete then what the article did. Perhaps an analogy of an infinitely precise ruler with every possible tick marked off (e.g. marking off 0.1 cm, 0.01, and so on for every real number), and how if in addition to labeling the ticks with lengths we also wanted to label them as first tick, second tick, third tick, it would be impossible. The basic gist of the proof from the article is given any proposal for a way to label the ticks on our ruler as first, second, third, ... covering all the ticks, there is a generic way to find a tick that is missed, so any such proposal must miss some ticks. So in a sense the real numbers are "bigger" than the integers, because if you try to match them up you will always have real numbers left over. Thus people say the real numbers are uncountable as there is no way to count them all.
This is in contrast to something like the integers where you could say 0 is the first one, 1 is second, -1 is third, 2 is fourth and so forth, eventually getting to all of them given infinite time with nothing left over.
I watched 'A Trip to Infinity' last night on Netflix and it touched on this part as well! Everything eventually spreads out so far away from everything else, that things burn out/die out in a very final kind of way. Its kind of sad, but also mostly irrelevant since we will long be gone by then(well, as far as we know!)
"God made the integers; all the rest is the work of Man." ~ Leopold Kronecker, when criticizing Cantor's work [0]. See also Stephen Hawking's Anthology by a similar name. [1]
[0] https://en.wikipedia.org/wiki/Leopold_Kronecker
[1] https://en.wikipedia.org/wiki/God_Created_the_Integers
[0] https://en.wikipedia.org/wiki/Leopold_Kronecker
[1] https://en.wikipedia.org/wiki/God_Created_the_Integers
I'd argue that the positive real numbers are existing/physical/God-made, while the negative numbers (even the negative integers) exist only to help the computations.
Yes. That's exactly the point! Same with imaginary numbers! And all sorts of other concepts. Such as different magnitudes of "infinity", which is what this post was about, what George Cantor studied, and the topic which prompted the quote! Everything other than the integers is just philosophy made up by consciousness.
You think e exists but 0 is only to help computations?
Related: Is Infinity Real? — https://www.quantamagazine.org/the-infinity-puzzle-solution-...
I came across this amusing question a while ago:
"How many computable numbers could a computer compute if a computer could computer all computable numbers?"
The correct answer (from HN comment section) appears to be 'countably many'.
Interestingly computable numbers don't go to infinity:
https://en.wikipedia.org/wiki/Computable_number#Not_computab...
> "While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable."
But they do get very very big, i.e. you can theoretically compute numbers so large that:
...even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe.
https://en.wikipedia.org/wiki/Graham%27s_number
"How many computable numbers could a computer compute if a computer could computer all computable numbers?"
The correct answer (from HN comment section) appears to be 'countably many'.
Interestingly computable numbers don't go to infinity:
https://en.wikipedia.org/wiki/Computable_number#Not_computab...
> "While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable."
But they do get very very big, i.e. you can theoretically compute numbers so large that:
...even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe.
https://en.wikipedia.org/wiki/Graham%27s_number
[deleted]
> Interestingly computable numbers don't go to infinity:
Computable numbers are an unbounded set, so in that sense they do go to infinity, just like the natural numbers go to infinity. They just aren't uncountable.
Others and myself have noted in other comments that the Cantor notion of cardinality is not all that interesting or useful compared to the concepts of intuitionalism and constructivism.
In particular, as noted in Wikipedia, "almost all real numbers are not computable", yet it is impossible (by definition) to actually produce or approximate one of these numbers through any sort of computational process. This sort of obvious nonsense is why mathematics has very very slowly been edging back from "Cantor's Paradise".
Computable numbers are an unbounded set, so in that sense they do go to infinity, just like the natural numbers go to infinity. They just aren't uncountable.
Others and myself have noted in other comments that the Cantor notion of cardinality is not all that interesting or useful compared to the concepts of intuitionalism and constructivism.
In particular, as noted in Wikipedia, "almost all real numbers are not computable", yet it is impossible (by definition) to actually produce or approximate one of these numbers through any sort of computational process. This sort of obvious nonsense is why mathematics has very very slowly been edging back from "Cantor's Paradise".
> In 1940 the famous logician Kurt Gödel proved that, under the commonly accepted rules of set theory, it’s impossible to prove that an infinity exists between that of the natural numbers and that of the reals. That might seem like a big step toward proving that the continuum hypothesis is true, but two decades later the mathematician Paul Cohen proved that it’s impossible to prove that such an infinity doesn’t exist! It turns out the continuum hypothesis can’t be proved one way or the other.
This, to me, is the exciting bit of the whole article.
There are questions that cannot be answered definitively. There is no way to say that it's true or false. Not simply that we haven't figured it out yet, but that no one, no matter how clever, will ever be able to prove the answer one way or another. If we meet an extraterrestrial alien civilization that is a million years of math theory ahead of us, they will not have an answer.
This, to me, is the exciting bit of the whole article.
There are questions that cannot be answered definitively. There is no way to say that it's true or false. Not simply that we haven't figured it out yet, but that no one, no matter how clever, will ever be able to prove the answer one way or another. If we meet an extraterrestrial alien civilization that is a million years of math theory ahead of us, they will not have an answer.
I think that a question has been answered definitely.
Continuum hypothesis is not provable within the theory along with the other axioms the the theory is derived from.
Similar to parallel postulate really
https://en.m.wikipedia.org/wiki/Parallel_postulate
Continuum hypothesis is not provable within the theory along with the other axioms the the theory is derived from.
Similar to parallel postulate really
https://en.m.wikipedia.org/wiki/Parallel_postulate
> along with the other axioms the the theory is derived from
Axioms are trivially provable in any system. Unless you mean prove them without using them, in which case you're actually talking about a system where they aren't axioms.
Axioms are trivially provable in any system. Unless you mean prove them without using them, in which case you're actually talking about a system where they aren't axioms.
If I remember correctly, this means that CH is independent of the other axioms of set theory. So in theory you could come up with another set of meaningful axioms (I.e. ones that produce and capture interesting mathematics), and in which you could prove CH.
So, an extraterrestrial civilization could in fact have an answer!
So, an extraterrestrial civilization could in fact have an answer!
The actual meaning of "continuum hypothesis can’t be proved one way or the other" is that the axioms we use in set theory are not sufficient to uniquely define what set is. They allow multiple possible interpretations. Under some of those interpretations, an infinity between N and R exists, under other interpretations it does not.
As a simple example, if I make the following set of axioms -- "each natural number is either big or small", "10 is a small number", "20 is a big number", "it cannot be simultaneously true that X is big and X+1 is small", -- then I can prove that 5 is small, or that 30 is big, but I can neither prove nor disprove that 15 is big. Because this set of axioms allow multiple possibilities for what "big" and "small" mean.
As a simple example, if I make the following set of axioms -- "each natural number is either big or small", "10 is a small number", "20 is a big number", "it cannot be simultaneously true that X is big and X+1 is small", -- then I can prove that 5 is small, or that 30 is big, but I can neither prove nor disprove that 15 is big. Because this set of axioms allow multiple possibilities for what "big" and "small" mean.
It's certainly cool, but don't make more of it than it is. It's merely pointing out that with a given set of axioms, there will be questions that cannot be answered. It's not a statement about the Real World Truth, but about the models we use.
The Cantor conception of "infinity" is bad and should be retired. It's a pure naval-gazing exercise that has set back practical mathematics significantly. The notion of the "continuum" is fundamentally flawed. Intuitionalism and constructivism are a much better framework for reasoning about mathematics.
The response to the idea that the natural numbers and the even numbers and the rational numbers and the computable numbers have the same "cardinality" should be "who cares". All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set.
It ends in the same trap as the Axiom of Choice, where you can produce lots of obviously incorrect results and remark about how amazing they are; similarly you can make nonsensical statements about things being true "almost everywhere" that are clearly actually true nowhere.
The response to the idea that the natural numbers and the even numbers and the rational numbers and the computable numbers have the same "cardinality" should be "who cares". All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set.
It ends in the same trap as the Axiom of Choice, where you can produce lots of obviously incorrect results and remark about how amazing they are; similarly you can make nonsensical statements about things being true "almost everywhere" that are clearly actually true nowhere.
> [Intuitionism] and constructivism are a much better framework for reasoning about mathematics.
Just for the sake of non-mathematicians, allow me to note here that despite intuitionism/constructivism being around for many decades, 99%+ of mathematics research done today is not, in fact, performed in these frameworks. These are just a curiosity that few working mathematicians actually care about. These approaches do have some certain philosophical benefits, but they have some extreme practical disadvantages that result in overwhelming majority of mathematicians rejecting the notion that these are “better frameworks”.
Your comment to me reads like saying that autogyro is a better framework for powered aviation than fixed wing or helicopters: regardless of your actual arguments in its favor (which may in fact be good), the fact that 99%+ of the industry disagrees is rather telling.
Just for the sake of non-mathematicians, allow me to note here that despite intuitionism/constructivism being around for many decades, 99%+ of mathematics research done today is not, in fact, performed in these frameworks. These are just a curiosity that few working mathematicians actually care about. These approaches do have some certain philosophical benefits, but they have some extreme practical disadvantages that result in overwhelming majority of mathematicians rejecting the notion that these are “better frameworks”.
Your comment to me reads like saying that autogyro is a better framework for powered aviation than fixed wing or helicopters: regardless of your actual arguments in its favor (which may in fact be good), the fact that 99%+ of the industry disagrees is rather telling.
In addition to this, I'd like to add that intuitionistic logic is consistent if and only if classical logic is. This follows from the Godel-Gentzen negative translation[0], which implies that for any contradiction in classical logic, you can get the same contradiction in intuitionistic logic more or less by adding "not not" before both sides of the contradiction. The same applies to the axiom of choice: set theory with choice is consistent if and only if set theory without choice is consistent[1].
This means that you don't get any safety by rejecting the law of excluded middle, nor by rejecting the axiom of choice. For this reason, I think intuitionistic logic is trading away a lot of power for basically no gain.
[0] https://en.wikipedia.org/wiki/G%C3%B6del%E2%80%93Gentzen_neg...
[1] https://en.wikipedia.org/wiki/Axiom_of_choice#Independence
This means that you don't get any safety by rejecting the law of excluded middle, nor by rejecting the axiom of choice. For this reason, I think intuitionistic logic is trading away a lot of power for basically no gain.
[0] https://en.wikipedia.org/wiki/G%C3%B6del%E2%80%93Gentzen_neg...
[1] https://en.wikipedia.org/wiki/Axiom_of_choice#Independence
> All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set.
But do you have something better? It’s not as if mathematicians immediately accepted this as the way forward. There was a long struggle accepting that lots of statements that are true about finite sets do not extend to infinite ones (examples: “adding an item to a set makes it larger”, “when summing a set of numbers, the result doesn’t depend on the order you do it”)
I think that you either have to accept this as the best way to treat infinite sets, or have to give up the notion of infinite sets, and that has its problems, too. For example, it would mean there’s a largest integer.
But do you have something better? It’s not as if mathematicians immediately accepted this as the way forward. There was a long struggle accepting that lots of statements that are true about finite sets do not extend to infinite ones (examples: “adding an item to a set makes it larger”, “when summing a set of numbers, the result doesn’t depend on the order you do it”)
I think that you either have to accept this as the best way to treat infinite sets, or have to give up the notion of infinite sets, and that has its problems, too. For example, it would mean there’s a largest integer.
We don't have to give up on infinite sets, but instead think of them as limiting processes instead of a complete object. Basically sacrifice the notion of "set" as a realized object. Just like sqrt(2) is not a rational number, but it is a computable number, which means that there is a computation that can yield an approximation that matches reality to whatever precision is requested.
Similarly the natural numbers are a process that can produce more natural numbers, and can do so for however large a limit you choose.
The problem of rearranging the elements of a set to get a different sum is handled here because you are looking at the limit of the sum, not the sum of the abstract set. Just as in classical math you have to make sure that to compute { sum_i=0^inf (-1)^i / i } that you are actually computing { lim N->inf sum_i=0^N (-1)^i / i }, with intuitionist math you can't help but do the latter, because the former statement makes no sense except as the latter.
The "adding an item to a set makes it larger" is only problematic if you're trying to define "larger" for non-terminating sets, and in addition trying to treat them as "sets" divorced from their underlying structure. The rationals and the integers may look the same to a set theorist, but they are clearly very different objects and it's not at all clear what additional power you get from introducing morphisms that don't preserve any underlying structure.
Similarly the natural numbers are a process that can produce more natural numbers, and can do so for however large a limit you choose.
The problem of rearranging the elements of a set to get a different sum is handled here because you are looking at the limit of the sum, not the sum of the abstract set. Just as in classical math you have to make sure that to compute { sum_i=0^inf (-1)^i / i } that you are actually computing { lim N->inf sum_i=0^N (-1)^i / i }, with intuitionist math you can't help but do the latter, because the former statement makes no sense except as the latter.
The "adding an item to a set makes it larger" is only problematic if you're trying to define "larger" for non-terminating sets, and in addition trying to treat them as "sets" divorced from their underlying structure. The rationals and the integers may look the same to a set theorist, but they are clearly very different objects and it's not at all clear what additional power you get from introducing morphisms that don't preserve any underlying structure.
Cantor's work on infinity and the diagonal argument was hugely important in mathematics, paving the way for important results like Godel's incompleteness theorems, the halting problem, the creation of modern set theory which allowed unifying effectively all of known mathematics into one theory, etc.
> It's a pure naval-gazing exercise
> The response to the idea [...] should be "who cares".
If anything sets back mathematics, it's when people have this kind of attitude towards the parts of math they find unintuitive.
> It's a pure naval-gazing exercise
> The response to the idea [...] should be "who cares".
If anything sets back mathematics, it's when people have this kind of attitude towards the parts of math they find unintuitive.
Mathematics is fine. Uncountable infinity is really only a problem in physics.
David Hilbert was a fan. Cantor's paradise he called it.
https://en.m.wikipedia.org/wiki/Cantor's_paradise
https://en.m.wikipedia.org/wiki/Cantor's_paradise
I'm not really familiar with math beyond linear algebra and googling was not very helpful; could you please give an example of an obviously incorrect result enabled by the axiom of choice? No need for detail, just the name of an example is good enough for me.
As far as axiom of choice goes, they were probably referring to https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Whether or not it is "wrong" is in the eye of the beholder. Why should infinite things be intuitive? After all even "real" things in the universe are highly unintuitive - e.g. quantum mechanics sounds obviously "wrong" at first glance.
Whether or not it is "wrong" is in the eye of the beholder. Why should infinite things be intuitive? After all even "real" things in the universe are highly unintuitive - e.g. quantum mechanics sounds obviously "wrong" at first glance.
Say Hi to Doron Z for me!
I just watched "A Trip to Infinity" [1] on Netflix last night and really enjoyed it. I'm not a mathematician by any means, but I was still able to follow it all. Highly recommended.
[1] https://www.netflix.com/title/81273453
[1] https://www.netflix.com/title/81273453
I liked this "music video" [1] inspired by the concepts involving proof of infinity
[1] https://www.youtube.com/watch?v=tNYfqklRehM
[1] https://www.youtube.com/watch?v=tNYfqklRehM
I second this recommendation. Excellent documentary.
I stopped believing in infinity. In the real, physical world. That makes much more sense to me and fits in much better with my understanding of the universe.
I agree. The practical intuitive way to think about “infinity” I use is to simply use a variable N that can be arbitrarily large. And then reason about f(N). It stops you from asking nonsenses like “what is infinity divided by infinity?” since you have to explicitly use N when asking the question. As in “what is f(N)/g(M) for arbitrarily large numbers N and M?”. Which gives you a straightforward way to answer the question.
To be fair, mathematics are not really concerned with the real, physical world. What’s interesting is that we can built axiomatic rules allowing us to properly define infinite sets and these constructions display surprising properties. It’s all purely abstract but it’s a fun thought exercise if you enjoy that kind of things. Plus it has practical applications sometimes which is good I guess.
Don't let the assholes here gaslight you into thinking that Finitism isn't a somewhat hetrodox but still respected field of math. A bunch of great mathematicians historically have felt this way, and a few great ones today still do.
Even if you're willing to accept the idea of cardinalities of infinity (and I think you ought to), I find that the more broad acceptance of 1. The axiom of choice (vs the axiom of determinancy, it's opposite) and 2. The law of excluded middle to be highly suspicious.
If you reject the axiom of choice, you're just in alternative but still correct math.
If you reject the law of excluded middle, you've ended up with intuitionistic or "constructive" logic, where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Notably, in intuitonistic/constructive logic, infinity is rejected until it can be "constructed", which also means that Cantors diagonalisation argument is not so naively accepted. While that diagonalisation itself was constructive, Other related and stronger theories from Cantor are not.
Even if you're willing to accept the idea of cardinalities of infinity (and I think you ought to), I find that the more broad acceptance of 1. The axiom of choice (vs the axiom of determinancy, it's opposite) and 2. The law of excluded middle to be highly suspicious.
If you reject the axiom of choice, you're just in alternative but still correct math.
If you reject the law of excluded middle, you've ended up with intuitionistic or "constructive" logic, where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Notably, in intuitonistic/constructive logic, infinity is rejected until it can be "constructed", which also means that Cantors diagonalisation argument is not so naively accepted. While that diagonalisation itself was constructive, Other related and stronger theories from Cantor are not.
I don't mind infinity in mathematics, it seems quite useful. I just don't think infinity exists in the real world.
To quote David Hilbert, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remain for the infinite to play is solely that of an idea."
To quote David Hilbert, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remain for the infinite to play is solely that of an idea."
Ahem. Let me show you a simple example.
Basic geometry. The diagonal of a right triangle.
End of the bullshit.
sqrt(2)
End of the bullshit.
Well if we're going to start deciding what things to believe in based on whether they're comfortable or not I am going to stop believing in traffic and the texture of coconuts.
For me, I stopped thinking of "infinity" meaning "bigger number than you can imagine" to just simply meaning "unlimited" or "never ending."
That way the term works equally well at different scales. There are infinitely many real numbers between 1 and 2. There are infinitely many natural numbers.
That way the term works equally well at different scales. There are infinitely many real numbers between 1 and 2. There are infinitely many natural numbers.
That's quite uncontroversial. But this is about math.
How do you reconcile the fact that the best tool we have, by far, to describe the "real, physical world" depends on certain notions of infinity?
> best tool we have
Math? Which physical laws/equations need infinity? Or do they just need "as x approaches infinity"?
But regardless, my only point was that in the real world everything is countable. The number of atoms in the sun, the distance to anywhere, the age of the universe, etc. Everything real is countable.
Math? Which physical laws/equations need infinity? Or do they just need "as x approaches infinity"?
But regardless, my only point was that in the real world everything is countable. The number of atoms in the sun, the distance to anywhere, the age of the universe, etc. Everything real is countable.
> Math? Which physical laws/equations need infinity?
Pretty much all of them are phrased in the language of differential equations.
Perhaps some can be salvaged through finitist approaches, but that seems far from certain or trivial.
Pretty much all of them are phrased in the language of differential equations.
Perhaps some can be salvaged through finitist approaches, but that seems far from certain or trivial.
[deleted]
Do you believe in 0?
Decent introduction to the concept of different infinities to the layperson.
However, this line:
> ... and some recent work has changed the way people think about the issue.
had me pretty excited I was about to read some brand new development. Alas, there is absolutely nothing new in this piece.
However, this line:
> ... and some recent work has changed the way people think about the issue.
had me pretty excited I was about to read some brand new development. Alas, there is absolutely nothing new in this piece.
For those who want some slightly more "sane" arguments against the foundation of mathamatics that is normally accepted, I highly recommend that folks read these wikipedia articles.
Basically, if your concerned with the epistemological justifications for the traditional way that infinite set theory is treated, it turns out you're far from alone! A lot of top mathematicians agree with you.
1. https://en.wikipedia.org/wiki/Intuitionism#Infinity
2. https://en.wikipedia.org/wiki/Intuitionistic_logic
3. https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...
3. https://en.wikipedia.org/wiki/Constructive_set_theory
4. https://en.wikipedia.org/wiki/Law_of_excluded_middle <---- (all of the above reject this naughty child, and rightfully so)
Basically, if your concerned with the epistemological justifications for the traditional way that infinite set theory is treated, it turns out you're far from alone! A lot of top mathematicians agree with you.
1. https://en.wikipedia.org/wiki/Intuitionism#Infinity
2. https://en.wikipedia.org/wiki/Intuitionistic_logic
3. https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...
3. https://en.wikipedia.org/wiki/Constructive_set_theory
4. https://en.wikipedia.org/wiki/Law_of_excluded_middle <---- (all of the above reject this naughty child, and rightfully so)
Infinite is a word like forever.
We have no problem understanding "ten minutes more than forever" makes no sense (other than as hyperbole), but a surprising number of people are unable to grasp that "one more than infinite" makes no sense.
Next up: The word unique, and why it similarly doesn't make sense to qualify it with phrases like "the most unique."
We have no problem understanding "ten minutes more than forever" makes no sense (other than as hyperbole), but a surprising number of people are unable to grasp that "one more than infinite" makes no sense.
Next up: The word unique, and why it similarly doesn't make sense to qualify it with phrases like "the most unique."
Why does "the most unique" not make sense? Object A can be different from all others in some way (so it is unique), but object B can be different in a much more extreme way. So B is "more unique" than A.
Unique means it's the only one like it. It can't be more only one than another thing. It can be more unusual or more interesting but it can't be more only one.
It’s totally fine to put gradations on “unique”. The “most unique” item in a set will have multiple aspects that are rare or one-of-kind. It may be the only green one, and also the only textured one and the only top-heavy one.
The more unique something is, the more it stands apart from other objects in its class.
The more unique something is, the more it stands apart from other objects in its class.
Unique means there's only one thing like it. You're advocating there's "more only one thing like this" of something than something else.
The word that takes qualifiers is "unusual." People commonly say "unique" when they mean unusual. One thing can definitely be more unusual than another thing.
The word that takes qualifiers is "unusual." People commonly say "unique" when they mean unusual. One thing can definitely be more unusual than another thing.
By their nature, all things are unique, depending on their framing. We make concessions because we don't all have infinite time.
Which is to say, you can't have 2 apples, you can only have one very specific, singularly unique apple and another very specific, singularly unique apple. If you get pedantic enough, nobody could ever have anything because the word invented to describe the first thing wouldn't technically describe the second thing perfectly (as they're not the same) so as to render speech useless.
Either accept that "unique" is a word for the poets and not the mathematicians, or accept that it's entirely useless as a descriptor.
Which is to say, you can't have 2 apples, you can only have one very specific, singularly unique apple and another very specific, singularly unique apple. If you get pedantic enough, nobody could ever have anything because the word invented to describe the first thing wouldn't technically describe the second thing perfectly (as they're not the same) so as to render speech useless.
Either accept that "unique" is a word for the poets and not the mathematicians, or accept that it's entirely useless as a descriptor.
I know, but still insist that the things that separate this unique item are themselves small or large.
If an item separates itself from the others on mutliple dimensions, then it is more unique than an item that differs only in one dimension.
Both of them are the only things like themselves. I have a collection of vases. Almost all of them are roughly the same size (15cm), shape (round; tapered), and color (clear or white). But there are two that are different: "A" is green instead of white, with a square base.
B is made from petrified buffalo dung, stands 1 meter tall, must be carefully balanced because it's so heavy on top (and skinny at the base), and is covered in velvet.
A is unique. B is definitely more unique.
If an item separates itself from the others on mutliple dimensions, then it is more unique than an item that differs only in one dimension.
Both of them are the only things like themselves. I have a collection of vases. Almost all of them are roughly the same size (15cm), shape (round; tapered), and color (clear or white). But there are two that are different: "A" is green instead of white, with a square base.
B is made from petrified buffalo dung, stands 1 meter tall, must be carefully balanced because it's so heavy on top (and skinny at the base), and is covered in velvet.
A is unique. B is definitely more unique.
All snowflakes are unique. Would a red snowflake be more unique? You might personally find the color axis more interesting, but it's just one of a huge number of axes along which the snowflakes are different. The red one is more interesting to you but it isn't actually "more unique". The same is true for your vases.
Most of the dictionaries I just Googled disagree with your assessment. It looks like unique has multiple definitions and while unique can definitely refer to something "being without a like or equal", other definitions include "unusual or special in some way" as well as something "rare and distinguished".
Like most arguments about words, it mostly comes down to context.
Like most arguments about words, it mostly comes down to context.
There are also dictionaries that say the definitions of infinite include "great or very great." I don't think most people at HN would accept those sloppy definitions of infinite any more than we should accept those sloppy definitions of unique.
Unique has a clear meaning. People frequently use the word unique in a sloppy manner to mean something different than unique, just as people frequently use the word infinite in a sloppy manner.
Unique has a clear meaning. People frequently use the word unique in a sloppy manner to mean something different than unique, just as people frequently use the word infinite in a sloppy manner.
Yes people, including those on HN, use the word infinite to mean very very large as opposed to the mathematical definition of infinite as a set that contains a proper subset with equal cardinality. Doing a Google search for uses of infinite on HN will reveal just as much, for example people complaining about websites with "infinite scrolling", or that the Fed has printed an infinite amount of money recently, or that software can be copied infinitely many times, or someone claiming that their friend has an infinite amount of memory.
In all these cases, they are using the word infinite to simply mean very very large.
In all these cases, they are using the word infinite to simply mean very very large.
Unique: being the only existing one of its type or, more generally, unusual, or special in some way
https://dictionary.cambridge.org/dictionary/english/unique
I think it depends on the context, but it does not seem an incorrect usage.
https://dictionary.cambridge.org/dictionary/english/unique
I think it depends on the context, but it does not seem an incorrect usage.
Omega + 1 is an original number that is larger than omega, which is infinite. This is well defined mathematics.
There are an infinite number of binary strings 0.xyzw... that are less than 1.0, and 1.0 is the first number after all of them. Infinity + 1.
There are an infinite number of binary strings 0.xyzw... that are less than 1.0, and 1.0 is the first number after all of them. Infinity + 1.
Negative numbers also made no sense for centuries and now we teach them to kids as meaning owning some amount of things.
The same could be said non-euclidean space (what is the use in that, right? or complex numbers) but both turned up to be useful in some contexts.
Same could be the case with fuzzy logic.
Cantor and the different infinite sizes is nonsense to some people but for some reason it is still there in the history of Mathematics. Maybe someone can explain better than me if it is useful, but there is a certain intuition to it that is interesting.
The same could be said non-euclidean space (what is the use in that, right? or complex numbers) but both turned up to be useful in some contexts.
Same could be the case with fuzzy logic.
Cantor and the different infinite sizes is nonsense to some people but for some reason it is still there in the history of Mathematics. Maybe someone can explain better than me if it is useful, but there is a certain intuition to it that is interesting.
Still, there are bigger and smaller infinities.
With ordinal transfinite numbers
infinity + 1
infinity + 2
Infinity × 2
Etc.
Are all nicely defined.
infinity + 1
infinity + 2
Infinity × 2
Etc.
Are all nicely defined.
yes, though, it is generally prefered to call it "omega" rather than "infinity".
Can't raise this topic without mentioning the Large Numbers Page:
http://www.mrob.com/pub/math/largenum.html
With this link being particularly relevant:
http://www.mrob.com/pub/math/largenum-10.html
http://www.mrob.com/pub/math/largenum.html
With this link being particularly relevant:
http://www.mrob.com/pub/math/largenum-10.html
I learned about Cantor and the concept of comparing infinities through George Gamow's excellent book, "One, Two, Three ... Infinity" many years ago. While most of the book's content on cosmology and atomic physics may be out of date (or now wrong), it's chapters on mathematics and Einstein's relativity are still a fun read.
So i've definitely heard of cantor's proof before, but one thing that struck me just now which i didn't really connect the dots on before, is how similar the diagonilaztion proof is to the proof of the halting problem.
Maybe its just coincidence, or maybe its not as similar as i think it is, but it does feel a bit surprising to me.
Maybe its just coincidence, or maybe its not as similar as i think it is, but it does feel a bit surprising to me.
Your observation is correct. This is pretty much the same argument.
https://en.wikipedia.org/wiki/Diagonal_argument
https://en.wikipedia.org/wiki/Diagonal_argument
Oh, I think this is the ideal place to share one of my favorite math blogs.
https://infinityplusonemath.wordpress.com/archive/
There is a section about 'infinity' and I think it is very fun to read.
https://infinityplusonemath.wordpress.com/archive/
There is a section about 'infinity' and I think it is very fun to read.
As a kid I have some religious friends and I was somewhat envious of them for not be able to feel something magical trough religion, but when I started questioning the universe and infinity stuff, oh boy, it was MY THING, I was so happy for not understand it.
How real is an imaginary number? Best to think of them as mathematical constructs and not get too hung up on them IMHO.
Imaginary numbers are more real than transcendental "real" numbers, and much more real than the unnameable reals, which are almost every real. It's easy to set up situation that points to any algebraic complex number.
> the unnameable reals, which are almost every real
Is there a definition you have in mind for "unnameable"? If you mean definable [0][1], then this is independent of ZFC.
The "standard" argument for this is flawed, and indeed there are models where every real is definable. [2]
[0] "x is definable if there exists a first order formula with one free variable P such that x is the unique real number with P(x) true" [1] https://en.wikipedia.org/wiki/Definable_real_number#Definabi... [2] https://mathoverflow.net/questions/44102/is-the-analysis-as-...
Is there a definition you have in mind for "unnameable"? If you mean definable [0][1], then this is independent of ZFC.
The "standard" argument for this is flawed, and indeed there are models where every real is definable. [2]
[0] "x is definable if there exists a first order formula with one free variable P such that x is the unique real number with P(x) true" [1] https://en.wikipedia.org/wiki/Definable_real_number#Definabi... [2] https://mathoverflow.net/questions/44102/is-the-analysis-as-...
Maybe the GP is referring to something like Chapter 5 of Chaitin's Meta Math!.
https://arxiv.org/pdf/math/0404335.pdf
https://arxiv.org/pdf/math/0404335.pdf
What does it mean for a number to be real (in the lower case sense of the word)?
[deleted]
“A Window on Infinity” chapter in the Beginning of Infinity is worth reading as an introduction.
[deleted]
No mention of the fact that infinity is not a number. Numeric operators only work on numbers.
I don't think that's a satisfactory answer for a layperson. Infinity is a size/quantity, for most people that usually means "number". The reason it's not a number is because it doesn't exist on the real number line, but again I don't think the helps people intuitively understand what's happening.
I think it does. A layperson scratches their head when trying to think about how to circle the square of "infinity + 1" or "infinity / infinity". Framing infinity as something outside of arithmetic is a starting point for a more interesting discussion of countability and types of infinities. Infinity is bigger than any number, so then how could it be a number?
Um, duh, you tip an 8 over. If it's not a number, why is it made out of a number? /s
There are numbers that aren't on the real number line. i, for one. Omega, for another.
It’s fairly easy to extend R with plus and minus infinity. It’s called the affinely extended real number system and you can extend the usual operations to somewhat work in it. It loses most of the interesting property of (R,+,x) but keeps enough for arithmetic to work mostly as expected.
Numeric operators only work on numbers. Operators can work on all sorts of mathematical objects.
Why would they need to mention this?
Does the universe even support infinity at absolute precision? If the data of an interim number used to calculate infinity, and that number could be represented by the using the smallest physical particle possible, said number would be limited to the size of the universe at a given time and the number of these hypothetical particles that could exist at said given time. In other words, the only things that can exist within the universe are things the universe is complex enough to support. There's nothing about the physical universe that suggests it can support anything remotely infinite. Infinity as a concept is handy for mathematics, but that's because we're not even close to being capable of calculating infinity to absolute precision (one reason why "Infinity" in programming languages is really just magic).
If infinity was an actual number (having a size), it would need to be extrauniversal unless it was possible for us to calculate it to the extent required, which has yet to be seen. The only candidate we have for proof that infinity exists is that the amount of time academics can spend pondering on infinity seems infinite.
If infinity was an actual number (having a size), it would need to be extrauniversal unless it was possible for us to calculate it to the extent required, which has yet to be seen. The only candidate we have for proof that infinity exists is that the amount of time academics can spend pondering on infinity seems infinite.
There’s also nothing about the physical universe to suggest that it is finite, or that it cannot contain infinities. It’s simply an open question.
> If infinity was an actual number (having a size), it would need to be extrauniversal
What does the universe or its physical extent have to do with the nature of numbers? A number is simply a concept, or idea, not necessarily a physical thing. We imagine things that aren’t physical all the time.
> If infinity was an actual number (having a size), it would need to be extrauniversal
What does the universe or its physical extent have to do with the nature of numbers? A number is simply a concept, or idea, not necessarily a physical thing. We imagine things that aren’t physical all the time.
> There’s also nothing about the physical universe to suggest that it is finite, or that it cannot contain infinities. It’s simply an open question.
The only evidence we have is that it's finite. Just because the universe expands does not mean that at any given time it is not finite.
> What does the universe or its physical extent have to do with the nature of numbers?
The only way to have scientific confidence in an idea is to test it. We can't calculate infinity because, infinity being something that isn't finite, every unit of information in our universe would have to be used to describe infinity. This doesn't work because there is never infinite information space in our universe at any given slice of time. Think of it this way; you can't take a modern video game and get it to perform exactly the same on a home computer from 1996 because it simply lacks the computing capacity. The only way it can work is to reduce certain aspects of the software to make it work at a much lesser capacity. There are no examples of any system that can describe another system more complex than itself with total accuracy. Thus, it makes no sense that a universe in which we are currently only able to describe through finite numbers would be able to support calculating what infinity as well as support the rest of its contents, if it can even do that at all.
This is why it's not at all accurate that a set of numbers can be "infinite." It only seems infinite because, for all intents and purposes, we don't have the capability to keep dividing a range of numbers forever. Even if we tried, the inevitable dissipation of heat energy would prevent us from doing so, that is if we don't simply run out of finite resources before then. If there is something that is indeed numerically infinite, we too would have to be infinite in order to make sense of it. We can't actually do that. That would be a contradiction. To illustrate this, go write some code that calculates every single number that exists in the "infinity" between two numbers. You won't be able to. Your software will fail because your computer doesn't support it. That is unless it has infinite bits.
A set where there's "infinite" numbers would more accurately be described as being indeterminate. The seeming "infiniteness" of one of these sets breaks down when you realize there's no way to even demonstrate that part of a set is infinite. From a conceptual standpoint, it's similar to how it might seem that a ball will fall straight down when you drop it, and it's generally useful to think of gravity in such a way, but that doesn't mean that it's actually so, just as believing that a set can be "infinite" might be useful, but the use of the term "infinite" for something finite like a set is incorrect.
So no, to others who think I'm being off topic. This is entirely on topic. A set being "infinite" gives you the wrong idea. At best, it defines a vector too large for humans or even human computers to find an end to. It's entirely virtual until proven otherwise.
The only evidence we have is that it's finite. Just because the universe expands does not mean that at any given time it is not finite.
> What does the universe or its physical extent have to do with the nature of numbers?
The only way to have scientific confidence in an idea is to test it. We can't calculate infinity because, infinity being something that isn't finite, every unit of information in our universe would have to be used to describe infinity. This doesn't work because there is never infinite information space in our universe at any given slice of time. Think of it this way; you can't take a modern video game and get it to perform exactly the same on a home computer from 1996 because it simply lacks the computing capacity. The only way it can work is to reduce certain aspects of the software to make it work at a much lesser capacity. There are no examples of any system that can describe another system more complex than itself with total accuracy. Thus, it makes no sense that a universe in which we are currently only able to describe through finite numbers would be able to support calculating what infinity as well as support the rest of its contents, if it can even do that at all.
This is why it's not at all accurate that a set of numbers can be "infinite." It only seems infinite because, for all intents and purposes, we don't have the capability to keep dividing a range of numbers forever. Even if we tried, the inevitable dissipation of heat energy would prevent us from doing so, that is if we don't simply run out of finite resources before then. If there is something that is indeed numerically infinite, we too would have to be infinite in order to make sense of it. We can't actually do that. That would be a contradiction. To illustrate this, go write some code that calculates every single number that exists in the "infinity" between two numbers. You won't be able to. Your software will fail because your computer doesn't support it. That is unless it has infinite bits.
A set where there's "infinite" numbers would more accurately be described as being indeterminate. The seeming "infiniteness" of one of these sets breaks down when you realize there's no way to even demonstrate that part of a set is infinite. From a conceptual standpoint, it's similar to how it might seem that a ball will fall straight down when you drop it, and it's generally useful to think of gravity in such a way, but that doesn't mean that it's actually so, just as believing that a set can be "infinite" might be useful, but the use of the term "infinite" for something finite like a set is incorrect.
So no, to others who think I'm being off topic. This is entirely on topic. A set being "infinite" gives you the wrong idea. At best, it defines a vector too large for humans or even human computers to find an end to. It's entirely virtual until proven otherwise.
I’ve read articles, seen interviews and read book by mathematicians and cosmologists on these issues. None of them agree with you as far as I can tell. Some cosmologists and physicists believe it’s likely the universe is finite, some believe it’s likely it’s infinite, but none I know of believe it’s a settled question.
Mathematicians perform calculations that include infinities routinely nowadays. They have sophisticated methods for it, and have discovered powerful tools and useful results from doing so.
Mathematicians perform calculations that include infinities routinely nowadays. They have sophisticated methods for it, and have discovered powerful tools and useful results from doing so.
> None of them agree with you as far as I can tell.
Oh really? How about Professor Hawking?
https://scitechdaily.com/constraining-the-multiverse-stephen...
> “We predict that our universe, on the largest scales, is reasonably smooth and globally finite. So it is not a fractal structure,” said Hawking.
> “We are not down to a single, unique universe, but our findings imply a significant reduction of the multiverse, to a much smaller range of possible universes,” said Hawking.
> I have never been a fan of the multiverse. If the scale of different universes in the multiverse is large or infinite the theory can’t be tested. [– Hawking]
Not that it particularly matters.
Cosmologists believe all sorts of things. Lots of cosmologists treat String Theory as more than a hypothesis despite how it hasn't yet panned out after decades and decades. Same goes for ideas like Dark Matter and Dark Energy, which are actually placeholders for some things we can't actually explain. That doesn't mean either of those things live up to their names or exist in any capacity. Hell, a substantial number of cosmologists profess that there's "got to be" alien life out there, despite no credible evidence what so ever. It's fine to hypothesize about all these things, but that doesn't make them even remotely fact, no matter what a cosmologist thinks. Science and mathematics aren't a majority vote.
> some believe it’s likely it’s infinite
They can believe that all they want, but that doesn't mean they're correct or that they have evidence. It's highly dubious whether it's even possible for us to verify the presence of infinitude.
> none I know of believe it’s a settled question.
That's because there are no settled questions. Any scientist or mathematician who professes that a question is settled immediately loses credibility by doing so.
The difference between infinitude and finitude is one can be demonstrated and the other has yet to be demonstrated. Guess which one hasn't been demonstrated. Just because mathematics can include a construct that allows describing an indeterminate range of numbers doesn't mean calling it "infinite" is accurate. An actual value that can even remotely be referred to as infinity hasn't been calculated and it's possible that it never will if the known universe doesn't even support calculating it. To do math with "infinity" is to use an indeterminate vector in a way that proves useful. It's no different than when someone uses a phrase like "this is taking forever."
Oh really? How about Professor Hawking?
https://scitechdaily.com/constraining-the-multiverse-stephen...
> “We predict that our universe, on the largest scales, is reasonably smooth and globally finite. So it is not a fractal structure,” said Hawking.
> “We are not down to a single, unique universe, but our findings imply a significant reduction of the multiverse, to a much smaller range of possible universes,” said Hawking.
> I have never been a fan of the multiverse. If the scale of different universes in the multiverse is large or infinite the theory can’t be tested. [– Hawking]
Not that it particularly matters.
Cosmologists believe all sorts of things. Lots of cosmologists treat String Theory as more than a hypothesis despite how it hasn't yet panned out after decades and decades. Same goes for ideas like Dark Matter and Dark Energy, which are actually placeholders for some things we can't actually explain. That doesn't mean either of those things live up to their names or exist in any capacity. Hell, a substantial number of cosmologists profess that there's "got to be" alien life out there, despite no credible evidence what so ever. It's fine to hypothesize about all these things, but that doesn't make them even remotely fact, no matter what a cosmologist thinks. Science and mathematics aren't a majority vote.
> some believe it’s likely it’s infinite
They can believe that all they want, but that doesn't mean they're correct or that they have evidence. It's highly dubious whether it's even possible for us to verify the presence of infinitude.
> none I know of believe it’s a settled question.
That's because there are no settled questions. Any scientist or mathematician who professes that a question is settled immediately loses credibility by doing so.
The difference between infinitude and finitude is one can be demonstrated and the other has yet to be demonstrated. Guess which one hasn't been demonstrated. Just because mathematics can include a construct that allows describing an indeterminate range of numbers doesn't mean calling it "infinite" is accurate. An actual value that can even remotely be referred to as infinity hasn't been calculated and it's possible that it never will if the known universe doesn't even support calculating it. To do math with "infinity" is to use an indeterminate vector in a way that proves useful. It's no different than when someone uses a phrase like "this is taking forever."
This is misunderstanding what the article is about. It does not make sense to talk about calculating a cardinal to infinite precision.
> this mysterious, complicated and important concept
For me it's not mysterious. I believe it's a fundamental phenomenon of the Multiverse and that the Universe was two or more Universes colliding which we call the 'big bang', a Universe among infinite amounts of Universes where multiple ones happen to collide all the time, sprouting new ones. How else can you explain our Universe spontaneously sprouting out of 'nothing'?
For me it's not mysterious. I believe it's a fundamental phenomenon of the Multiverse and that the Universe was two or more Universes colliding which we call the 'big bang', a Universe among infinite amounts of Universes where multiple ones happen to collide all the time, sprouting new ones. How else can you explain our Universe spontaneously sprouting out of 'nothing'?
> How else can you explain our Universe spontaneously sprouting out of 'nothing'?
It's turtles all the way down. Where did this multiverse come from? Some hyperverses colliding together? Where did the hyperverses come from? etc. etc. At some point, something seems to have sprouted from nothing.
Either that, or "nothing" and "something" are fundamentally the same thing. The distinction of the two is an illusion of the human mind.
It's turtles all the way down. Where did this multiverse come from? Some hyperverses colliding together? Where did the hyperverses come from? etc. etc. At some point, something seems to have sprouted from nothing.
Either that, or "nothing" and "something" are fundamentally the same thing. The distinction of the two is an illusion of the human mind.
>How else can you explain our Universe spontaneously sprouting out of 'nothing'?
You explained nothing. Where did the multiverse come from then.. did it sprout out of nothing?
You explained nothing. Where did the multiverse come from then.. did it sprout out of nothing?
Yes but Infinity explains that. It just goes on and on...there is no beginning.
If that's what you need multiverse for, then it's perfectly compatible with our current understanding of the big bang to claim that it is without any beginning and that it was expanding and expanding from smaller and smaller thing forever.
Both theories equally non falsifiable and make use of infinity.
Both theories equally non falsifiable and make use of infinity.
Infinity is infinitely big. Conversely, it also must be infinitely small, almost nonexistent. Because, in order to be infinite, it must contain everything, including all concepts that are contradictory.
So it ends up in the eternal philosophizing of many esoteric schools and religions: Infinity is the ultimate balance, tranquility, that is totally inert and irrelevant to outside because it contains all the contradictions inside itself in a balancing, canceling-out fashion.
But...
Infinity must also be the total opposite of that in order to be infinite - it must be the total opposite of infinitely balanced too. Again canceling out any description and defying identification.
So it all begins and ends in mystery.
Then again, it also must not be mysterious in order to be truly infinite, so...
So it ends up in the eternal philosophizing of many esoteric schools and religions: Infinity is the ultimate balance, tranquility, that is totally inert and irrelevant to outside because it contains all the contradictions inside itself in a balancing, canceling-out fashion.
But...
Infinity must also be the total opposite of that in order to be infinite - it must be the total opposite of infinitely balanced too. Again canceling out any description and defying identification.
So it all begins and ends in mystery.
Then again, it also must not be mysterious in order to be truly infinite, so...
"Because, in order to be infinite, it must contain everything, including all concepts that are contradictory."
No, this is a common misconception but it is false. Consider the set of all even integers. It is infinite, but no matter how long you search you will never find 3. There is no sense in which "infinity" entails "all inclusive".
You can define "the set that contains everything", but it is also not terribly interesting that it contains "contradictions". Clearly the set that contains everything contains the propositions "2 is even" and "2 is not even", but... so what? All that implies is that contradictory claims exist, which is not even slightly profound. Prove contradictory claims are somehow both true and now you're cooking with philosophical gas, but the mere fact they can be defined is uninteresting. That is literally nothing more than the observation that both false and true statements exist.
No, this is a common misconception but it is false. Consider the set of all even integers. It is infinite, but no matter how long you search you will never find 3. There is no sense in which "infinity" entails "all inclusive".
You can define "the set that contains everything", but it is also not terribly interesting that it contains "contradictions". Clearly the set that contains everything contains the propositions "2 is even" and "2 is not even", but... so what? All that implies is that contradictory claims exist, which is not even slightly profound. Prove contradictory claims are somehow both true and now you're cooking with philosophical gas, but the mere fact they can be defined is uninteresting. That is literally nothing more than the observation that both false and true statements exist.
> Consider the set of all even integers
Nope, don't: Mathematical approaches to infinity do not work. Because all mathematical approaches are limited. And the infinite cannot be expressed by using the finite.
Infinity must contain everything that exists and their antithesis in it. If even one thing is missing, the infinity won't be infinite.
> but it is also not terribly interesting that it contains "contradictions"
It's terribly interesting. Because:
> the propositions "2 is even
The infinity must contain the antithesis of a proposition. The antithesis of 2. The antithesis of even. The antithesis of everything involved in making that proposition. If it doesn't, then its not infinite because it is missing something.
Nope, don't: Mathematical approaches to infinity do not work. Because all mathematical approaches are limited. And the infinite cannot be expressed by using the finite.
Infinity must contain everything that exists and their antithesis in it. If even one thing is missing, the infinity won't be infinite.
> but it is also not terribly interesting that it contains "contradictions"
It's terribly interesting. Because:
> the propositions "2 is even
The infinity must contain the antithesis of a proposition. The antithesis of 2. The antithesis of even. The antithesis of everything involved in making that proposition. If it doesn't, then its not infinite because it is missing something.
So, you have your idiosyncratic definition of infinity. You are welcome to it. But you are making the very common mistake of thinking you can play games with definitions and thereby somehow affect the real world. Along with failing to change anything with your definitions, you also sever yourself from any ability to communicate with anyone else in the world using a more conventional definition.
I see this sort of approach a lot. It may give you a lot of warm fuzzy mystical feelings, but it's sterile. It can't go anywhere beyond that. You've locked yourself in a tiny little box with your fuzzy feelings and mixed up definitions, and you think you've got a hold of something amazing, but you're just... locked in a tiny box, with nowhere to go, with nothing allowed to come in and affect you and no ability to put anything out into the world.
What's more, if you came out into the greater world with the rest of us, you'd find that your content-free mystical feelings are really just a counterfeit of the wonder that is available in the real world. More complicated, more substantial, more interesting, and as far as I can see, an infinite (heh) vista of things to discover and communicate, rather than a tiny little concept posing as a large one.
Throw away your attachment to your idiosyncratic definition and join the rest of us on the real voyage of discovery. It may hurt a bit at first, and it certainly requires more work, but the pay off is worth it.
I see this sort of approach a lot. It may give you a lot of warm fuzzy mystical feelings, but it's sterile. It can't go anywhere beyond that. You've locked yourself in a tiny little box with your fuzzy feelings and mixed up definitions, and you think you've got a hold of something amazing, but you're just... locked in a tiny box, with nowhere to go, with nothing allowed to come in and affect you and no ability to put anything out into the world.
What's more, if you came out into the greater world with the rest of us, you'd find that your content-free mystical feelings are really just a counterfeit of the wonder that is available in the real world. More complicated, more substantial, more interesting, and as far as I can see, an infinite (heh) vista of things to discover and communicate, rather than a tiny little concept posing as a large one.
Throw away your attachment to your idiosyncratic definition and join the rest of us on the real voyage of discovery. It may hurt a bit at first, and it certainly requires more work, but the pay off is worth it.
> Because, in order to be infinite, it must contain everything
This is wrong (or at least, using a poor choice of definition of infinite). I don't get why people think this?
This is wrong (or at least, using a poor choice of definition of infinite). I don't get why people think this?
> I don't get why people think this?
Because, otherwise at least one thing will be missing from infinity, and it wont be infinite in that direction. And that's not the mathematical sense of directions, physics vectors etc. All of them are always limited and they cannot describe infinity. Infinity must be infinite in every way.
Its the opposite: A lot of people get stuck at a certain point because they try to describe infinity using mathematical concepts or thinking. Mathematics, which is a framework that is limited in specific ways in order for humans to be able to understand it and calculate through it.
Because, otherwise at least one thing will be missing from infinity, and it wont be infinite in that direction. And that's not the mathematical sense of directions, physics vectors etc. All of them are always limited and they cannot describe infinity. Infinity must be infinite in every way.
Its the opposite: A lot of people get stuck at a certain point because they try to describe infinity using mathematical concepts or thinking. Mathematics, which is a framework that is limited in specific ways in order for humans to be able to understand it and calculate through it.
Ok, so you are just using an IMO mostly useless definition of the term. Fine. That mostly just means you aren’t talking about the same thing as everyone else.
Though, what do you call something which isn’t finite then?
Or perhaps you define “finite” as “not ‘infinite’” where ‘infinite’ is being used as you use it?
In which case, that also makes “finite” mostly useless imo,
and I question what you would use to describe something which not only has limits of some kind, but is specifically limited in terms of how much stuff it has, in addition to being limited in terms of what stuff, and specifically, where the “how much” is limited as being less than some natural number.
Though, what do you call something which isn’t finite then?
Or perhaps you define “finite” as “not ‘infinite’” where ‘infinite’ is being used as you use it?
In which case, that also makes “finite” mostly useless imo,
and I question what you would use to describe something which not only has limits of some kind, but is specifically limited in terms of how much stuff it has, in addition to being limited in terms of what stuff, and specifically, where the “how much” is limited as being less than some natural number.
> mostly useless
Its a philosophical approach. Not a mathematical. It does not have to be one.
> what you would use to describe something which not only has limits of some kind, but is specifically limited in terms of how much stuff it has, in addition to being limited in terms of what stuff, and specifically, where the “how much” is limited as being less than some natural number.
That's a given set of specific limitations. Quite a mathematical concept. Like 'the set of natural numbers up to a given, large number'. It only makes sense within the given limitations and definitions, a precise mathematical use case. Otherwise if you dive into its details, like how an infinite amount of fractional numbers inhabit even that given large set of natural numbers, things go out of hand there too.
Its a philosophical approach. Not a mathematical. It does not have to be one.
> what you would use to describe something which not only has limits of some kind, but is specifically limited in terms of how much stuff it has, in addition to being limited in terms of what stuff, and specifically, where the “how much” is limited as being less than some natural number.
That's a given set of specific limitations. Quite a mathematical concept. Like 'the set of natural numbers up to a given, large number'. It only makes sense within the given limitations and definitions, a precise mathematical use case. Otherwise if you dive into its details, like how an infinite amount of fractional numbers inhabit even that given large set of natural numbers, things go out of hand there too.
I don’t think it really is a particularly math-specific concept. I mean, it is mathematical in the same way that “how many fingers am I holding up?” is mathematical, but it isn’t specific to any particular axiomatization of something, and is instead expressed in terms of general intuitive notions.
If I have a box, I might ask, “is there some number n such that there are fewer than n protons in this box?”. Well, really, the question would be phrased “are there finitely many protons in the box”, but I’m avoiding the word “finite” because you use the word “infinite” differently.
The question of “are there ‘un-split-able ones’ ?” goes way back, and is a question about the world, not about math considered abstractly. Aiui, it was considered philosophically. A question as to whether there is a limit to how much things can be split into smaller parts.
The question of “Does this thing have a limit of ‘how much’/‘how many’ ?“ is not a question which concerns only mathematicians, but which concerns anyone who seeks to understand the world.
If I have a box, I might ask, “is there some number n such that there are fewer than n protons in this box?”. Well, really, the question would be phrased “are there finitely many protons in the box”, but I’m avoiding the word “finite” because you use the word “infinite” differently.
The question of “are there ‘un-split-able ones’ ?” goes way back, and is a question about the world, not about math considered abstractly. Aiui, it was considered philosophically. A question as to whether there is a limit to how much things can be split into smaller parts.
The question of “Does this thing have a limit of ‘how much’/‘how many’ ?“ is not a question which concerns only mathematicians, but which concerns anyone who seeks to understand the world.
Countable numbers (1, 2, 3, ...) are infinite, in the sense that they never end. However, the information complexity of a program to generate them is finite and only a few bytes long. Information complexity of generating digits of π is considerably more complex but still very finite.
So what you really want is the infinity of infinite information complexity, but we haven't discovered such a thing yet.
So what you really want is the infinity of infinite information complexity, but we haven't discovered such a thing yet.
> Countable numbers (1, 2, 3, ...) are infinite
They mathematically are. They are not truly infinite. Because, they are countable and define-able as a set. They are infinite in only one direction.
> what you really want is the infinity of infinite information complexity
Still falls short. The infinity must contain the antithesis of information. And complexity.
They mathematically are. They are not truly infinite. Because, they are countable and define-able as a set. They are infinite in only one direction.
> what you really want is the infinity of infinite information complexity
Still falls short. The infinity must contain the antithesis of information. And complexity.
Looking at your comments, it seems you are merely defining infinity to what you want it to be.
Which is fine. It is, after all, what the mathematicians did.
But then taking your definition and insisting other definitions are wrong - that's problematic.
Which is fine. It is, after all, what the mathematicians did.
But then taking your definition and insisting other definitions are wrong - that's problematic.
Only complaint is the attempt to build intuition came very close to making an incorrect statement:
> But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers: Here it’s the numbers 4, 5 and 6. But there will always be infinitely many real numbers between them, numbers like 3.001, 3.01, π, 4.01023, 5.666… and so on.
> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation.
The alluded property (the set of real numbers is dense) is not related to cardinality, in the sense that the rational numbers are dense but also countable. But I appreciate that the author is trying to motivate the remaining explanation about diagonalization, which can be a tricky topic for beginners.