# 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 ...
• 20.3k

### Replication crisis in mathematics

Mathematics does have its own version of the replicability problem, but for various reasons it is not as severe as in some scientific literature. A good example is the classification of finite simple ...

### 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 ...
• 79.5k
Accepted

### "Strange" proofs of existence theorems

There are probabilistic proofs of existence. Do they fall into one of your three categories? For example, prove the existence of a real number that is normal in all bases: To do it, we show that "...
• 40.6k

### 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 ...
• 116k
Accepted

### Did Euler prove theorems by example?

There's some evidence that precisely the opposite can be said: that Euler is aware of the fallacies of proving theorems by example (of course, this does not necessarily mean he has never used it). One ...

### Did Euler prove theorems by example?

"Proof by example" is a technique used by Euclid, who often proved results that hold e.g. for n integers in a typical case, say for 3 integers, as well as by Diophantus, who had to choose values for ...

### "Strange" proofs of existence theorems

There is a famous proof of the existence of two irrational numbers $a$ and $b$ such that $a^b$ is rational. The proof considers two cases: $\sqrt{2}^\sqrt{2}$ is irrational, or it is rational. In ...
• 23.1k
Accepted

### How can the simply typed lambda calculus be Turing-incomplete, yet stronger than second-order logic?

The simply-typed $\lambda$-calculus is not stronger than second-order logic. The simply-typed $\lambda$-calculus has: product types $A \times B$, with corresponding term formers (pairing and ...
• 47.8k

### How is it possible for PA+¬Con(PA) to be consistent?

As is being discussed in the comments, the non-standard proofs in non-standard models of $\mathsf{PA}$ are not trustworthy (in that they're not sound). I'll give a sort of 'toy model' illustrating the ...
• 10.4k

### Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

Regarding your second question, Apéry's amazing formula $$\zeta(3) = {5\over 2} \sum_{n\ge1} {(-1)^{n-1} \over n^3 {2n \choose n}}$$ has inspired the search for analogous formulas for other zeta ...
• 79.5k
Accepted

### Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

The proof of irrationality of $\displaystyle\sum_{n=0}^{+\infty}\frac1{F_n}$ (where $F_n$ is the $n$-th Fibonacci number) by RIchard André-Jeannin is an adaptation of the original Apery's proof of the ...
• 3,801

### "Strange" proofs of existence theorems

This is the strangest existence proof I know; it is a nonconstructive proof that there exists a proof of a certain statement. In other words, we prove the statement by proving that a proof exists. I'...
• 42.4k
As Akiva Weinberger conjectured, this depends on the implementation. Indeed, $0=0$ is a sentence of this type, i.e. $0=0$ is equivalent to the claim that there is a proof of $0=0$ that is shorter than ...