If I have a keyboard, then for a relatively small dataset, such as my saved passwords, I prefer a text file over any database. Lookup workflow: decrypt file (with, say, gpg), find what I want (with, say, grep), delete decrypted file. (Not safe if you don't use disk encryption!) Update workflow: decrypt file, edit file, encrypt file, commit encrypted file to local git repo, git push to backup storage.
Set theorists have located the "end" for all practical and most impractical purposes. Let M be the minimal countable transitive model of ZFC. Declare a real to be useful if and only if it is in M.
If you're giving the book away, why not release the source (LaTeX, Word, whatever) in addition to your favorite subset of {PDF,EPUB,...}? Then readers who want a different format have a better chance of being able to produce it themselves.
Every formal axiom system, whether about sets or anything else, can only prove that N of Busy Beavers halt, for some (not-too-large in practice) finite number N. If you just want to maximize N, right now your best bet is set theory plus a very strong large cardinal axiom, like I0.
Perhaps simply because it is older, set theory is way ahead of its competitors in developing a hierarchy of extremely strong (but apparently consistent) axioms to supplement its base theory ZFC.
Nice overall; my only complaint is that you raised the question "Are LTC log returns normally distributed?" but only gave an uninterpreted graphical comparsion. I was hoping for a test statistic...
Indeed, the source code only considers tax brackets, the personal exemption, and the standard deduction. I've read of an increase of $650 in the per-child tax credit combined with the loss of the per-dependent personal exemption and an increase in the standard deduction. If your last $4333 of annual income is going to be taxed at rate of 15%, then a $650 credit is just as valuable as a $4333 deduction or exemption and thus slightly more valuable than the current $4050 personal exemption. Thus, contrary to this calculator, middle-class families with lots of kids will not get big tax increases.
Yes. Perhaps the voting currency should be hours, not dollars. Make the first vote free. But N-1 more votes costs you N^2-1 hours from your "registered community service" account maintained by the government. No overdrafts allowed: the hours must be credited before they can be debited.
The Carnot cycle is a poor model for steam turbine electric power plants. Rankine cycle efficiency of ~40% is the more
relevant theoretical upper limit.
In the real world, Hamkins' program would search forever, never outputting anything. In the context of a "nonstandard" version of the line of natural numbers, "finite" no longer means what you think it means. In this context, Hamkin's program does output something after a "finite" number of steps, and Hamkins can make the program output whatever he wants by changing the meaning of "finite" instead of changing the program.
I think your watch is slow and you think mine is slow in the following sense. For each of many events, I record their time t and position x while you, moving relative to me at constant velocity, observe a different time T and position X. Between two events with the same X-value, such as ticks of your pocket watch, the difference in t-times is larger than the difference in T-times by a factor gamma. Conversely, between two events with the same x-position, such as ticks of my pocket watch, the difference in T-times is larger than the difference in t-times by that same factor gamma.
There is no contradiction because events with identical x-positions do not have identical X-positions.
In contrast (but not contradiction) with your experience, as an MIT undergraduate math major applying for PhD programs, my advisor warned me that the math dept. doesn't admit their own graduates. I applied anyway and indeed was not admitted (which was fine since I was admitted to UW-Madison, where I most wanted to go).
Theorems 1 and 3 are finitary and easier to physically interpret. As for Theorem 2,
"With probably p a random foo has property bar" intends no interpretation of "random foo"; it merely claims that the set of foo's with property bar has measure p with respect to some probability measure. In this case, the probability measure is the Bernoulli measure.
There's no relation to his incompleteness theorems. Gödel proved that every universe V of sets that satisfies the (ZFC) axioms has a subuniverse L that satisfies both the axioms and the continuum hypothesis (CH). L is constructed by transfinite recursion. At each stage of the construction, the only new sets you admit are the definable subsets of the set of all sets you admitted in earlier stages.
Unfortunately, proving that L satisfies ZFC and CH is a very delicate and technical matter. On the other hand, it can be fun because it's extremely meta. One of the steps of the argument is proving that L is the L of L. I learned the proof from a graduate textbook chapter titled "defining definability."
The man's escape strategy described in the post was discovered in 1952 by Besicovitch. It has a kind of discontinuity because, quoting the post, "There are 2 perpendicular paths. Choose the one closest to the center, or if the two paths are equally close, then either one is fine." In simple terms, the man will eventually make a wrong turn (with probability 1) unless he can instantaneously measure distances with perfect accuracy.
Surprisingly, for every continuous man strategy, there is a continuous lion strategy that can catch the man by time T where T is the disc radius divided by the lion's speed.
Restricting to continuous strategies in some other lion-man games actually leads to other paradoxes such as both lion and man having a "winning" strategy. The shallow resolution of the paradox is that two such winning strategy cannot actually be played against each other.
The deeper resolution is that even continuous strategies can be unphysical if they allow for information to travel at infinite speed (e.g., if the man is modelled as knowing the lion's current speed and velocity, special relativity notwithstanding). I'm not aware a proof in the literature, but presumably continuity of strategies plus an information speed limit will avoid the above paradoxes. (Continuous-time game theory is still very immature compared to discrete-time game theory.)
https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charge...