Problem 1 asks to determine (with brief justifications) the truth value of the following statements about integers:

1. is False. To show this we provide a counterexample: Specific integers such that For example,

2. is False. To show this we need to exhibit for each integer an integer such that For example, Note that, although is a fixed integer once we know we are not giving a fixed value of that serves as a simultaneous counterexample for all values of

3. is True. To show this we exhibit for each integer a specific integer such that For example: Note that, although is a fixed integer once we know we are not giving a fixed value of that works simultaneously for all

4. is True. To show this, we exhibit specific values of such that For example:

Problem 2 asks to show by contradiction that no integer can be both odd and even. Here is the proof: Suppose otherwise, i.e., there is an integer, let’s call it such that is both odd and even. This means that there are integers such that (since is odd) and (since is even).

Then we have that or But this is impossible, since 1 is not divisible by 2. We have reached a contradiction, and therefore our assumption that there is such an integer ought to be false. This means that no integer can be both odd and even, which is what we wanted to show.

Note that we have not shown that every integer is either odd or even. We will use mathematical induction to do this.

Problem 3 asks for symbolic formulas stating Goldbach’s conjecture and the twin primes conjecture (both are famous open problems in number theory).

Goldbach’s conjecture asserts that every even integer larger than 2 is sum of two primes:

Here, is the formula asserting that is even, namely, and is the formula (given in the quiz) asserting that is prime. Note we had to add existential quantifiers in order to be able to refer to the two prime numbers that add up to

The twin primes conjecture asserts that there are infinitely many primes such that is also prime.

The difficulty here is in saying “there are infinitely many,” since the quantifier only allows us to mention one integer at a time, and writing something of infinite length such as is not allowed.

We follow the suggestion given in the quiz, and represent “there are infinitely many with [some property]” by saying “for all there is a larger with [some property].”

43.614000-116.202000

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I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

Consider any club subset of $\kappa $. Check that it has order type $\kappa>\lambda $, and that its $\lambda $th element (in its increasing enumeration) has cofinality $\lambda $.

A very nice introduction to this area is MR0891258(88g:03084). Simpson, Stephen G. Unprovable theorems and fast-growing functions. In Logic and combinatorics (Arcata, Calif., 1985), 359–394, Contemp. Math., 65, Amer. Math. Soc., Providence, RI, 1987. Simpson describes the paper as inspired by the question of whether there could be "a comprehensive, self […]

There are continuum many (i.e., $|\mathbb R|$) such functions. First of all, there are only $|\mathbb R|$ many continuous functions, so this is an upper bound. On the other hand, for any real $r$, $f(x)=x+r$ satisfies the requrements, so there are at least $|\mathbb R|$ many such functions.

I'm posting an answer based on Asaf's comments. The following reference addresses this question to some extent: MR0525577 (80g:01021). Dauben, Joseph Warren. Georg Cantor. His mathematics and philosophy of the infinite. Harvard University Press, Cambridge, Mass.-London, 1979. xii+404 pp. ISBN: 0-674-34871-0. Reprinted: Princeton University Press, P […]