# 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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### Does the finitary proof of the consistency of relevant PA shows that first order PA is irrelevant?

Relevance logic takes a closer look at the implication operation in first-order logic. It suggests that implications such as: p and not p -> q cannot hold; in ordinary English, an example of this ...
2answers
174 views

### Can we expand the notion of truth in Godels incompleteness theorem?

Godels incompleteness theorem, which really should be called the undecidability theorem given that the paper of Godels which this theorem is taken from is named 'On formally undecidable propositions ...
3answers
493 views

### What is the connection between provability logic & Gödel's first incompleteness theorem?

Provability logic is a modal logic that interprets the modal operator of K as provability and an additional axiom derived from Löb's theorem. Now the SEP shows that it's possible to derive Gödel's ...
1answer
211 views

### Is there a finitistic set theory, and if there is, is it provably consistent?

ZFC is the mainstream set theory. It has an axiom of infinity which claims that there is at least one infinite set. Now suppose like Aristotle, we object and say that there are no actually infinites ...
1answer
713 views

### What is the impact of paraconsistency on Gödel's theorem?

Russell's paradox forced a restriction of the natural abstraction principle (that every predicate determines a set) so that set theory could be consistent; the standard one being ZF. However ...
2answers
624 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 ...
1answer
574 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 ...
1answer
273 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 ...
3answers
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### 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 ...
3answers
206 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 ...
5answers
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### 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, ...
3answers
319 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 ...
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 ...
3answers
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 ...
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 ...
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 ...
3answers
295 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 ...
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 ...
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 ...
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 ...