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Tagged with peano-axioms incompleteness
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Godel's incompleteness theorem: Question about effective axiomatization
I am studying Godel incompleteness theorems and I am struggling with the definition of effective axiomatization.
From Wikipedia:
A formal system is said to be effectively axiomatized (also called
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Peano Arithmetic can prove any finite subset of its axioms is consistent
Timothy Chow writes in a MathOverflow answer
[...] here is a classical fact: for any finite subset of the axioms of PA (remember that PA contains an axiom schema and hence has infinitely many axioms),...
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Contradiction and Godel's incompleteness theorems
If T is a recursively axiomatizable formal system containing peano arithmetic and is able to carry out the proof for the Godel's incompleteness theorems (so according to Wikipedia includes primitive ...
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Is Gödels second incompleteness theorem provable within peano arithmetic?
All following notation and assumptions follow Gödel's Theorems and Zermelo's Axioms by Halbeisen and Krapf.
Exercise 11.4 c) states "Conclude that the Second Incompleteness Theorem is provable ...
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PA + "(PA + this axiom) is consistent"
By Gödel's second incompleteness theorem, no sufficiently powerful formal system can prove its own consistency.
I was wondering what happens if one tries to manually append an axiom stating a formal ...
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Precise statement of Gödel's Incompleteness Theorems [duplicate]
I have seen the following statements of Gödel's Incompleteness Theorems:
Gödel's First Incompleteness Theorem (v1) If $T$ is a recursively axiomatized consistent theory extending PA, then $T$ is ...
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Confusion about $\mathsf{PA}$'s self-provable consistency sentences
Edit: This long question was basically answered by a quick comment! I'll accept an answer if someone posts one, but it's basically answered already.
Background:
In Peter Smith's Introduction to Gödel'...
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Infinite statements from finite axioms
I want to know if a given finite subset of axioms of PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite ...
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is arithmetic finitely consistent? [duplicate]
Let's take PA1( First order axioms of peano arithmetic ) for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest ...
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Gödels incompleteness theorem false for natural numbers
Do I understand it correctly that in an axiomatic system that includes the natual numbers defined by using Peano's axiom, where the 9th axiom (induction) is formulated using 2nd-order logic, then
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Are the natural numbers definable in ZFC-Inf
While reading up on the incompleteness theorems on this page, I came upon the statement that the incompletess theorems hold in ZFC-Inf. According to the page ZFC-Inf is 'proof-theoretically equivalent'...
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If Goldbach's conjecture G is undecidable in PA, then can we prove $\mathbb N\models G$?
Suppose Goldbach's conjecture $G$ is undecidable in first-order Peano arithmetic, $\sf{PA}$. That would mean there are models in which $G$ is true and other in which it is false. But intuitively, this ...
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Are there Nonstandard Models of Arithmetic that don't Add Additional Axioms?
Apologies if this is an elementary question that should have been obvious to me. I am learning about these topics very much from the perspective of an outside hobbyist, and am not a wizard of logic.
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Gödel's second incompleteness theorem and Consistency.
According to Gödel's second incompleteness theorem, no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. As I understand it, this result contributed to spark a ...
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What is the mathematical definition of "standard arithmetic/standard natural numbers"?
As a consequence of Godel's incompleteness theorem, no axiomatic system can be the definition of standard natural numbers because any axiomatic system of arithmetic will always be satisfied by non-...
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