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Your question seems to concern the issue of the computability of solutions of computable functions, and the larger context for such a question is the subject known as computable analysis. Carl Mumme …

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 …

The other examples are great! Meanwhile, there is a translation between this question and the eventual counterexamples question. Namely, For any property $Q(m)$ with eventual counterexamples, the p …

In model theory, an object is algebraic in a structure $M$ if it satisfies a property that only finitely many other objects in $M$ exhibit, where by "property" here we mean one that is expressible in …

If a theory has only one model, then of course it is complete, in the sense that any statement or its negation is a consequence of the theory, because either it holds in the unique model or it doesn' …

This article contains the following statements. Describing the normality property, Bailey explains that "in the familiar base 10 decimal number system, any single digit of a normal number occ …

Such a kind of question is the central concern of the field of mathematics known as Reverse Mathematics. The goal of the subject is to find exactly which axioms are needed to prove which theorems, ove …

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 …

The MRDP solution of Hilbert's 10th problem establishes that the integer solution sets $\{\, n\, |\, \exists\vec n\, p(n,\vec n)=0\}$ of diophantine equations $p(n,\vec n)=0$ are exactly the computabl …

You inquire about comparing your algorithm to a given recursive algorithm, but the more fundamental question would seem to be how good is your algorithm just by itself? There are numerous ways to me …

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 …

The integers can indeed be defined in the rational field, but not in the real field. $\newcommand\Q{\mathbb{Q}}\newcommand\Z{\mathbb{Z}}\newcommand\R{\mathbb{R}}$ The question can be made precise by i …

The other answers and comments are fascinating, particularly about the irrationality measure, but allow me to give a little more information along the lines of Mark Sapir's answer by mentioning that t …

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, …

Allow me to reinterpret your question as the inquiry How can abstract infinitary constructions inform us about the finite? To my mind, this is the troubling or at least surprising possibility at the …