No, the question here is not "are there statements in _this axiom system_ that cannot be proved by this axiom system" (what incompleteness theorem says), but "are there statements about _the positive integers_ that cannot be proved by this axiom system". In other words, is this system of axioms equivalent to the axioms that define positive integers.
I had the exact same idea, right down to using ratings to improve the random match, so this is pretty awesome in that I get to see how this will play out for free.
The main idea here is that it's a behaviorial approach to optimizing mate-matching. The way it works now (e.g., on okcupid) is that you fill out a profile and answer some questions. You expose some signals in your profile which you think are correlated with what makes you a good mate (interests, income, etc.), and go about finding such signals in other people's profiles. The match % is also such a signal. However,
1. People lie, or at least exaggerate the truth/selectively filter
2. They might not lie but simply be wrong: they think they want x but actually want y
3. The match % is a completely made up algorithm. I mean, it kind of makes sense that people who agree on things they care the most about would make good couples, but unless you measure this in a scientific way (i.e., controlled experiments), you have no idea.
4. Even assuming that you can actually get a good signal from the profile, it is drowned out by the noise. Suppose that some girl and dude are perfect for each other. What are the chances that he will find her profile and that she will read his message?
The solution to 1-3 is to measure the thing you want to optimize, instead of optimizing based on intuition. The solution to 4 is to set people using the thing that you measured. So, after each blind date, you ask the participants, "would you go on a date with this person again?" Imagine being okcupid and being able to say, here is a person I think you'd be interested in dating, and I can say with 95% confidence that you will have a good time. If they can get to that stage, they'd be able to make good money.
There are some issues with this, of course. You could argue that the thing they should measure is whether the date leads to a happy long-term relationship, but that's pretty much infeasible for various reasons. Also, as a dating website, they'd be perversely incentivized to keep you single forever. Anyway, I am super interested to see how this does.