Questions about Gödel's incompleteness theorems and related topics.
Questions about Gödel's theorems and related topics.
Questions tagged [incompleteness]
868 questions
8
votes
1 answer
Gödelian incompleteness; Smullyan's Puzzle
I am currently doing exercises on the Gödelian theorems; and we are confronted with the introductory puzzle of R. Smullyan's book, which is as follows:
Suppose we have a machine which prints strings over the alphabet $p, n, (, ), \neg$. The norm of…
trin
- 81
7
votes
2 answers
Is Goedel term (in incomleteness theorem) both true and unproveable?
In Goedel incompleteness theorem is Goedel term both true and unproveable, or just unproveable and truth neutral?
Can we add Goedel term to the theory as axiom and get new theory?
Can we add Goedel term negation to the theory and get a new theory?…
Suzan Cioc
- 931
7
votes
4 answers
The Penrose–Lucas argument
I was looking at the Penrose–Lucas argument as discussed on Wikipedia. It states:
In 1931, the mathematician and logician Kurt Gödel proved that any
effectively generated theory capable of expressing elementary
arithmetic cannot be both…
Clinton
- 503
6
votes
5 answers
Gödel says: countable proofs, uncountable conjectures?
I thought I understood Gödel's Incompleteness Theorem to say:
Starting from ZF, there only a countable number of proofs you can write
The number of possible conjectures is uncountable.
Thus, there is a conjecture S such that you can't write down…
user2469
6
votes
2 answers
Does Gödel's incompleteness theorem contradict itself?
I have problems understanding Gödel's incompleteness theorem. I presume I have a misunderstanding of some phrase or I have to look closer at the meaning of some detail.
Gödel's second incompleteness theorem states that in a system which is free of…
Daniel S.
- 530
5
votes
1 answer
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 theory with recursively enumerable theorems can prove…
oozer
- 61
5
votes
0 answers
Must there be a consistent "math"?
The incompleteness theorem is true for sufficient complex systems. Is it known if there must be at least one such sufficient complex system which is consistent or could it be that every such system is inconsistent?
Thank you very much
Kevin Meier
- 1,535
4
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1 answer
About Gödel's Incompleteness Theorem
A short time ago, I've been thinking about what statements in Number Theory are true but not provable. I've seen the proof of the Incompleteness Theorem (in the Gödel's works) and he gave an example of one statements which is true but not provable,…
Alexei0709
- 1,184
3
votes
2 answers
Talking about Gödel's incompleteness theorems...
I will give a talk about Gödel's incompleteness theorems to a group of people consisting of undergraduates of mathematics, graduates of informatics and etc (they are not really familiar with the subject in question). I am an undergraduate student of…
Pranasas
- 1,370
3
votes
1 answer
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 false,' I'm just struggling to link the two…
staplegun
- 63
3
votes
1 answer
Theory $T$ that is consistent, such that $ T + \mathop{Con}(T)$ is inconsistent
Is it possible for a theory $T$ to be consistent, but for $T$ + $\mathop{Con(T)}$ to be inconsistent?
Demi
- 329
3
votes
2 answers
Is there more than one underivable, true theorem in a formal system under godel?
I've been reading Douglas Hofstadter's excellent book "Godel, Escher, Bach: An eternal golden braid" and I think I understand the proof for Godel's incompleteness theory, but I still have a couple of questions.
My first question is this:
One can…
Kevin Milner
- 145
2
votes
0 answers
Is a proof through undecidability contradictory?
I'm reading Fermat's Last Theorem by Simon Singh and in it he writes the following:
... if Fermat's Last Theorem turned out to be undecidable, then this would imply that it must be true. The reason is as follows. The Last Theorem says there are no…
Griffin Staples
- 21
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2
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1 answer
Godel's incompleteness theorem and proof by contradiction
This is a naive question and I haven't actually read the paper itself (I've read this). But from my understanding he demonstrated that it is possible to encode the statement "This statement cannot be proved" using Gödel numbers...
I've also heard…
profPlum
- 337
2
votes
0 answers
Math's inconsistency: can we prove that some statements are False and only False?
I just started to study Gödel's Incompleteness Theorems and have a doubt.
I know that, if Math is inconsistent, than every statement can be proven to be True.
This happens because if we can prove that (a) $X$ is True and that (b) $X$ is False, than…
