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Tagged with peano-axioms axioms
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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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Finite axiomatization of EFA
According to the paper Fragments of Peano's Arithmetic and the MRDP theorem (Section 6), elementary function arithmetic (EFA) is finitely axiomatizable.
Is there a known explicite finite ...
user1343124
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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),...
- 4,886
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Is there a problem if I don't use $0$ in Peano arithmetic?
Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema.
$\forall x \ (0 \neq S ( x ))$
$\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
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Allegedly: the existence of a natural number and successors does not imply, without the Axiom of Infinity, the existence of an infinite set.
The Claim:
From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:
[T]he existence of natural numbers and the fact that given a natural ...
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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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In the axiomatic treatment of natural numbers, can we define what a natural number is?
In his book Number Systems and the Foundations of Analysis, Elliot Mendelson first defines a Peano system (p. 53). He then goes on to prove that two arbitrary Peano systems are isomorphic — they are ...
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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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Peano axiom of induction with "no junk"
In this Wikipedia treatment of Peano Axioms, if you go down to the first picture you'll see a circle of dominoes and a straight line of dominoes:
The caption says the straight line of dominoes
The ...
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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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How can one understand the axiomatization of numbers and their operations?
I've recently taken an interest in arithmetic and how it can be axiomatized. I haven't been reading very deeply about some of the known axioms I've come across, like Peano axioms, which by my ...
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Fifth Peano axiom — Properties of the natural numbers
This question is kind of a follow-up question to this. I am also using Terence Tao's book and I still struggle to understand why the fifth Peano axiom is valid.
Tao defines the fifth axiom in the ...
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Can exponentiation be defined in Robison's Q by use of the exponential Diophantine equation?
Robinson's Q is an axiomatization of arithmetic which only defines addition and multiplication, and does not have the axiom schema of induction (unlike Peano axioms).
I was wondering, given ...
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Extending empty set + adjunction to interpret PA
Let N = empty set + adjunction. N interprets Q.1 Q + induction yields PA.
Does N + epsilon-induction interpret PA? If so:
Are they mutually interpretable, sententially equivalent, and/or bi-...
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Operation 'Referencing' In Abstract Algebra
I recently started an Abstract Algebra course so I may not know all of the 'necessary' vocabulary to fully express my question, but here is my attempt. First I need to provide some background thinking....
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