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".
112
questions
2
votes
1
answer
155
views
Can Internal Set Theory provide a complete system of arithmetic?
Internal Set Theory (IST) is a conservative extension of ZFC that adds three axioms that serve to define a predicate standard such that all numbers are either standard or not. There are finitely many ...
- 27.3k
-3
votes
1
answer
150
views
Is this a paradox or a mistake? [closed]
Zhang Hong recently asked "is there a paradox lurking in Godel's 1931 incompleteness proof" (paraphrase)? This can be answered in two ways: First, by proving quite generally that there are ...
- 11
6
votes
5
answers
5k
views
Is there a paradox in the proof of Godel's incompleteness theorem?
The Gödel Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness ...
- 115
1
vote
0
answers
61
views
Was Gödel actually convinced that his ontological proof was correct?
The proof is obviously logically valid, but it is as obvious that it isn't logically sound.
For instance, the second axiom states that ¬P(φ) ⟺ P(¬φ), take φ(x) ⟺ x is a male human being. Then either ...
- 127
1
vote
0
answers
92
views
Completeness in finished system
Gödel's incompleteness theorems addresse formal axiomatic théories.
Incompleteness of arithmetic of natural numbers is an example.
My question is if a theory regarding a finite class of numbers cannot ...
- 345
6
votes
2
answers
435
views
How does Gödel’s encoding of mathematical statements into natural numbers enable self-referential propositions?
As part of his proof for the first incompleteness theorem, Gödel encoded mathematical expressions into unique numbers. These were used to construct statements exhibiting self-referentiality, such as ...
- 1,138
2
votes
3
answers
457
views
Is it a problem for arithmetic or our representation (or both) that there is incompleteness?
Is this a settled (as much as it can be) philosophical area? I feel like I understand that there will always be incompleteness for a finite set of axioms trying to capture all of arithmetic. But I ...
- 2,942
4
votes
3
answers
130
views
Do Gödel sentences (even indirectly) assert only their own unprovability?
Sometimes the basic Gödel sentence is said to mean something like, "This sentence is unprovable in system F." Perhaps more correctly, it is sometimes said to mean something like, "There ...
- 16.7k
1
vote
1
answer
52
views
Do the incompleteness theorems need the provability predicate to be expressed, or can they be expressed via just ⊢?
In his "Epistemic Set Theory," William Reinhardt says:
It is the purpose of this paper to formulate axioms for Gödel's modal operator B for provability (see [3], [8]) in the context of set ...
- 16.7k
0
votes
2
answers
129
views
Is Gödel's incompleteness theorem based on premisses bearing a contradiction? [closed]
In a 7-page article, Chaim Perelman provides an argument supporting the idea that Gödel's premises for the incompleteness theorem bear a contradiction.
Has this ever been refuted? If yes, how? Does ...
-2
votes
1
answer
101
views
Please evaluate my argument about incompleteness theorem and first cause
Here is my argument:
One of the incompleteness theorems is
“If a system is noncontradiction, it is incomplete”
Incomplete means that there are propositions that are true but cannot be proven.
The ...
- 59
7
votes
12
answers
4k
views
Can you mathematically prove the existence of God?
So I came across this video (https://www.youtube.com/watch?v=z0hxb5UVaNE), which claims to prove the existence of God using math.
I then searched and found stuff like this: mathematician Kurt Gödel's ...
- 79
0
votes
1
answer
81
views
A technical question about the limitation of z of "jointing together" or "zus(x,y)" in Gödel Arithmetization
I am recently reading Professor Carnap's Logical Syntax of Language.
In p.61 D18.1., the limitation of z is not greater than: pot [prim (sum[lng(x), lng(y)]), sum(x,y)].
Remarks: z is the series-...
3
votes
4
answers
264
views
Prerequistes for mathematical logic
I have a working knowledge of calculus and linear algebra. But when I pick up books on mathematical logic (for example the ones listed in the logic study guide by Peter Smith), they often use ...
- 43
2
votes
1
answer
97
views
Kant's commentary on the faculty of judgment: did he anticipate things like incompleteness/halting/truth-undefinability?
First, to cite the (Meiklejohn) version of the argument:
If understanding in general be defined as the faculty of laws or rules, the faculty of judgement may be termed the faculty of subsumption ...
- 16.7k