10 questions linked to/from Understanding Gödel's Incompleteness Theorem
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### Can someone explain Gödel's incompleteness theorems in layman terms?

Can you explain it in a fathomable way at high school level?
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### 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 ...
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### 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 ...
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### 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?
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### 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'...
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### 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 ...
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### 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 ...
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### 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 ...