Questions tagged [goedel]

Questions related to the work of Kurt Gödel. Please mind the spelling of his last name: "Gödel". If you cannot or don't know how to create the "ö", you might also write his name as "Goedel". In all cases, please avoid "Godel". If you want to create a (hyper)link to, say, a Wikipedia entry, you might have to manually change the "ö" to "%F6".

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5
votes
2answers
615 views

Is Gödel's incompleteness theorem still valid if one uses a higher-order logic?

Gödel's incompleteness theorem is wholly formal (in my understanding), and relies on a proof system that I assume is first-order. Does it make any difference to the theorem if higher-order logic is ...
8
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1answer
566 views

In Gödels Incompleteness theorem what is the notion of truth?

The entry on Gödels Incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for ...
4
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1answer
272 views

What happens when we drop the condition that proofs are finite strings of inferences in Gödel's incompleteness theorem?

Gödel's incompleteness theorem shows that there are sentences that are undecideable, that is they nor their negation can be proved. This theorem operates purely syntactically or formally, that it ...
29
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3answers
3k views

How is Gödel's incompleteness theorem interpreted in intuitionistic logic?

Classically, one sets up an axiomatic system with a formal deduction system & an interpretation in a model. Generally it is sound, that is: a formally deduced theorem is also true when interpreted ...
3
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3answers
203 views

Can we add to PA a new predicate T such that for every sentence A of the old vocabulary the new theory proves T(Godel numeric number of A) iff A

I am new to logic but I believe this is not a difficult problem, yet I am still soo confused, and the reason for that is because there are so many gaps in my knowledge or maybe I have overlooked so ...
12
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5answers
2k views

Gödel's theorem and God

I have seen it argued that Gödel's Incompleteness Theorems have implications regarding the existence of God. Arguments for the existence of God run mostly along the lines: "Because of Gödel's Theorem, ...
1
vote
3answers
312 views

Was Gödel the first person to bring up that truth always exceeds the grasp of proof?

Was Gödel the first person to pose and solve this question in mathematics? In the larger philosophical debate, has this question been posed before? Say by Plato or Aristotle? One could interpret for ...
7
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6answers
1k views

What's the big deal with Gödel's second incompleteness theorem?

Edit: My question is specifically about Gödel's second incompleteness theorem. I get the significance of his first incompleteness theorem, which is of course completely amazing. According to the ...
3
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2answers
2k views

Relation of Gödel's incompleteness theorems and Karl Popper falsification

Falsifiability is considered a positive (and often essential) quality of a hypothesis because it means that the hypothesis is testable by empirical experiment and thus conforms to the standards of ...
34
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3answers
11k views

Did Russell understand Gödel's incompleteness theorems?

Russell was active in philosophy (although no longer in math) for many years after the Gödel's 1931 publication. Gödel's paper were not obscure, and Russell would have been aware of their effect on ...
8
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5answers
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What are the philosophical implications of the Halting Problem?

In a great answer, a community member gave the following proof sketch that the halting problem is undecidable: Proof that the halting problem is undecidable. If there were a computable procedure to ...
3
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3answers
293 views

Do Gödel's incompleteness theorems support the idea that the examination of a 'system' should only be undertaken to arrive at the inconsistency?

Roughly, Gödel demonstrated that in a logical system, that contains a model or arithmetic, there are statements which may be true, but are unprovable within the system. If a statement is not ...
16
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1answer
3k views

When it is correct to use Tarski's undefinability theorem versus Gödel's incompleteness theorem?

Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say ...
29
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4answers
12k views

What are the philosophical implications of Gödel's First Incompleteness Theorem?

Gödel's First Incompleteness Theorem states Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any ...
74
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8answers
11k views

Is Kurt Gödel's Incompleteness Theorem a “cheap trick”?

I found a throw-away critique of Kurt Gödel's Incompleteness Theorem in an essay about Deconstruction: The basic enterprise of contemporary literary criticism is actually quite simple. It is based on ...