Questions about Gödel's incompleteness theorems and related topics.

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2
votes
1answer
135 views

Why does the existence of independent statements not prove completeness?

I've read before that, by the Principle of Explosion, if a theory is inconsistent, then absolutely any statement can be proven within it. Obviously, there are statements which are independent of ZFC ...
4
votes
1answer
158 views

Does robinson arithmetic satisfy modal logic's “axiom 4”?

Does Robinson arithmetic prove the theorem "if sigma is provable then 'sigma is provable' is provable' for a fixed sentence sigma? It's clear to me that you can get a primitive recursive function f ...
2
votes
1answer
174 views

How to express Con(PA) as a first-order statement?

I read from somewhere that Fact 1. PA, which refers to the first-order version, is not finitely axiomatizable. At the same time, the second incompleteness theorem says that there is no proof in ...
1
vote
1answer
160 views

Quantitative version of Godel's incompleteness theorem

Let $A$ be a list of axioms which we assume to be sound (for example, PA or ZFC). Godel's incompleteness theorems imply that if we add only finitely many (true) axioms to $A$, the new list $B$ will ...
12
votes
1answer
374 views

What's the theory in which incompleteness of PA is proved?

Maybe this is a dumb question, but I have to admit that it is not really clear to me what the theory is, in which incompleteness of PA and stronger theories is proved. The texts I did study so far are ...
6
votes
1answer
217 views

Best known theory for proving statements about natural numbers

Gödel's incompleteness theorems imply that there is no consistent theory that can be effectively generated and contains all true statements about the natural numbers. Well, what known consistent ...
5
votes
3answers
882 views

Prove Gödel's incompleteness theorem using halting problem

How can you prove Gödel's incompleteness theorem from the halting problem? Is it really possible to prove the full theorem? If so, what are the differences between original proof and proof by ...
6
votes
4answers
640 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) ...
2
votes
1answer
172 views

Construction of a sequence of theorems with increasing and unbounded “difficulty”?

Let's define the "difficulty" of a theorem as the logarithm of the size of its shortest proof divided by the logarithm of the size of the theorem itself. For example, if a theorem has difficulty less ...
9
votes
2answers
256 views

How to show the existence of an infinite set of independent undecidable sentences?

How to show the existence of an infinite set of independent undecidable sentences? By "independent" I mean that no implication between any two elements is provable. A finite set that satisfies the ...
1
vote
1answer
288 views

Why is Godel's first theorem not a proof from the inside?

Why is Godel's first theorem not a proof for the truth of the so called undecidable proposition? You may say it's a proof from the outside, but if not all proofs from the outside be formalized inside ...
2
votes
1answer
416 views

Is Robinson Arithmetic complete and not-complete?

Is Robinson Arithmetic complete in the sense of Gödels completeness theorem? And is Robinson Arithmetic incomplete in the sense of Gödels first incompleteness theorem? If RA is both it would be a good ...
7
votes
1answer
373 views

Can Robinson's Q prove Presburger arithmetic consistent?

I made an assertion in What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? that Q has higher consistency strength than Pres, Presburger arithmetic; ...