# Tag Info

Accepted

### What does it mean to suspect that two conjectures are logically equivalent?

First of all, in practice when we say "Conjecture A is equivalent to Conjecture B," what we mean is "We have a proof that Conjecture A is true iff Conjecture B is true." We can have such a proof ...
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### What does it mean to suspect that two conjectures are logically equivalent?

Noah Schweber's answer is in some sense the "right" answer, and I would have said something similar if he hadn't beaten me to it. However, I think that it's worth pointing out that reverse ...
• 78.6k

### Is there a database for tracking the dependencies of mathematical theorems?

The reverse mathematics zoo, founded by Damir Dzhafarov and with recent development by Eric Astor, aims to be a database showing the relations and dependencies of mathematical theorems as described in ...

### What does it mean to suspect that two conjectures are logically equivalent?

One way to say rigorously what it means for two true theorems to be equivalent is to find a meaningful way to generalize both of them to statements that aren't always true, and then prove that they're ...
• 115k

### Why is this new result such a big deal?

The statement in question, frequently denoted $\mathsf{RT}^2_2$ in the context of reverse mathematics, is the instance of the infinite Ramsey theorem for unordered pairs and two colors. Specifically, ...
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### Is there a database for tracking the dependencies of mathematical theorems?

It's important to distinguish between two types of "dependencies." The cleanest type of dependency is the kind that is studied in reverse mathematics: Theorem T depends on Axiom A if T is actually ...
• 78.6k
Accepted

### What is known about the relationship between Fermat's last theorem and Peano Arithmetic?

The main reference for this topic is Angus Macintyre's appendix to Chapter 1 ("The Impact of Gödel's Incompleteness Theorems on Mathematics") of Kurt Gödel and the Foundations of Mathematics: ...
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### Comprehension axiom that helps in the opposite direction

David Belanger's work is relevant. The principle $\mathsf{WKL_0}$ (= "Every infinite subtree of $2^{<\omega}$ has an infinite path") is not literally a comprehension principle, so its ...
• 19.2k

### Is Monsky's theorem provable in $\mathsf{RCA}_0$?

This isn't an answer to your question because I have no idea whether this argument can be carried out in $RCA_0$, but this fact doesn't appear to be mentioned anywhere easily accessible through ...
• 115k
TL;DR: A most basic property of $\mathbb{R}$ is that it is not countable, which is surprisingly hard to prove (namely far beyond the Big Five you mention), as explored in [1, 2, 3]. The longer version....
Let me strengthen Emil's excellent answer to a finer reverse mathematical result. Recall that $\mathsf{RCA}_0^\star=\mathsf{EA}+\mathsf{I}\Delta^0_1+\Delta^0_1\textsf{-CA}$ and \$\mathsf{WKL}_0^\star=\...