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Here are a few areas of overlap for those research topics. Computable model theory is a nice overlap of computability theory and algebra, since one is looking at the nature of computably effective pr …

Here is an easy way to see it. Let $A$ assert that if PA is inconsistent, the smallest $k$ for which $\Sigma_k$ induction is inconsistent is a multiple of $3$. Let $B$ make the similar assertion tha …

I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately …

I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he …

The two notions are incomparable. To see that the first notion does not imply the second, let's construct a set $S$ with asymptotic density $0$, but with infinite chromatic number. We place infinitely …

Consider the arithmetic progression game, a two-player game of perfect information, in which the players take turns playing natural numbers, or finite sets of natural numbers, all distinct, and the fi …

To start things off, here is a simple observation: the set $S$ is contained in the rational interval $\mathbb{Q}\cap[\frac 12,1]$, the rational numbers $\frac ab$ where $0<a\leq b\leq 2a$. The reaso …

Picking up on the phrase "arithmetic information" in your question, let me give a brief answer coming from logic, although I recognize that this is likely not the answer for which you are looking. L …

Nice question, Erin. Here is one quick easy thing to say. If $\pi$ and $e$ disagree in infinitely many digits, then there are continuum many choices of the particular subset of those digits to swap, …

I once heard Harvey Friedman suggest that the set of prime-differences, that is, the set of all natural numbers $n$ for which there are primes $p,q$ with $p-q=n$, as a possible candidate for all we kn …

I had pointed out earlier that the question as asked has a negative answer, in light of the counterexample provided by my answer to your previous question. Theorem. There is a coloring of ordinals wi …

The answer is no; this generalization is inconsistent, even with just two colors, and with $F=\{\omega,\omega^2\}$ of size two. Theorem. There is a coloring of ordinals with two colors, such that for …

Your proposed treatment of having machines use binary input with the alphabet $\{0,1\}$, where $0$ counts as a blank symbol (so that the input is padded with infinitely many additional $0$s, is not Tu …

I note that the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$ all have nonzero solutions in the ordinals, since for any positive natural number $n$ we have $$17^n+\omega^n=\omega^n, $$ wh …

The ordinals below $\omega^2$ are exactly those of the form $\omega\cdot n+k$ for natural numbers $n$ and $k$. Thus, these are the ordinals having two digits in base $\omega$, and counting to $\omega^ …