522 views

### What other unprovable theorems are there? [duplicate]

Gödel's incompleteness theorem presents us with the possibility of having theorems that are neither provable nor disprovable in a given axiomatic set. Already we have the continuum hypothesis which ...
21k views

### What are some counter-intuitive results in mathematics that involve only finite objects?

There are many counter-intuitive results in mathematics, some of which are listed here. However, most of these theorems involve infinite objects and one can argue that the reason these results seem ...
10k views

### Examples of problems that are easier in the infinite case than in the finite case.

I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can't think of any good ones for now, but I'll be sure to add some when I do.
18k views

### What is a simple example of an unprovable statement?

Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements. I've seen a simple natural language ...
22k views

### Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
4k views

### What is this operator called?

If $x \cdot 2 = x + x$ and $x \cdot 3 = x + x + x$ and $x^2 = x \cdot x$ and $x^3 = x \cdot x \cdot x$ Is there an operator $\oplus$ such that: $x \oplus 2 = x^x$ and $x \oplus 3 = {x^{x^x}}$? ...
1k views

### How can Busy beaver($10 \uparrow \uparrow 10$) have no provable upper bound?

This wikipedia article claims that the number of steps for a $10 \uparrow \uparrow 10$ state (halting) Turing Machine to halt has no provable upper bound: "... in the context of ordinary ...
198 views

### How can you come to the truth of a statement without proving it?

I was reading a bit about Gödel's incompleteness theorems. I haven't took the time to really study it, but I'm very curious about statements like these: In other words, if our axioms are consistent ...
101 views

### Can PA prove very fast growing functions to be total?

The Goodstein-sequence is a total function, but PA cannot prove this. Is this true for any other function with growth rate at least $f_{\epsilon_0}$ or are there functions growing at least as fast ...
For example, if we want to prove that $a^2+b^2\ge 2ab$ for all $a,b\in\mathbb{R}$, we will start from something which is true (axiom or something that is already proved). In this case we will use fact ...