Linked Questions

129
votes
13answers
77k views

Can someone explain Gödel's incompleteness theorems in layman terms?

Can you explain it in a fathomable way at high school level?
61
votes
16answers
4k views

Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious. Can you give me examples of conjectures/theories that seemed true but through rigorous ...
25
votes
1answer
2k views

Computability viewpoint of Godel/Rosser's incompleteness theorem

How would the Godel/Rosser incompleteness theorems look like from a computability viewpoint? Often people present the incompleteness theorems as concerning arithmetic, but some people such as Scott ...
9
votes
1answer
4k views

Are Real Numbers axioms a consistent or complete system?

Do we know if the axioms of the real numbers are consistent, complete or neither of both? And if so, is it a consequence of Godel's theorem or of something else?
6
votes
3answers
993 views

A naive inquiry of Godel's incompleteness--or why does mathematics need proofs of unprovability?

My question stems from reading Swetz, 1994 (mostly excerpts from the journal Mathematics Teacher) and Berlinski, 2005 (a popular book on 10 most important mathematical breakthroughs in history). 1) I'...
3
votes
3answers
368 views

Why can PA + $\neg G_{PA}$ be consistent?

Wikipedia and other sources claim that $PA +\neg G_{PA}$ can be consistent, where $\neg G_{PA}$ is the Gödel statement for PA. So what is the error in my reasoning? $G_{PA}$ = "$G_{PA}$ is ...
3
votes
2answers
494 views

Gödel's incompleteness theorems: where to learn? Is there a straightforward relation between the two?

What would be a good textbook or paper to learn the proofs of the two Gödel's incompleteness theorems from? I would prefer it to be as close to the original proofs as possible. I have not tried to ...
2
votes
2answers
309 views

Consequences of Incompleteness.

Suppose that I'm a working mathematician that has just proved a Theorem, say, in Number Theory. Does Gödel's Incompleteness Theorem imply that I can't know for sure if there exists a proof of the ...
0
votes
1answer
369 views

How does Godel use diagonalization to prove the 1st incompleteness theorem?

I'm looking for an intuitive explanation of this without too much jargon as I am new to set theory. I understand Cantor's diagonal proof as well as the basic idea of 'this statement cannot be proved ...
0
votes
1answer
231 views

What does First Godel's Incompleteness theorem mean?

I am terribly confused what really "Incomplete" mean in terms of the Godel's theorem. Does it mean there are some theorems that are definable in First order theory of natural numbers and true but not ...