Questions tagged [peano-axioms]

For questions on Peano axioms, a set of axioms for the natural numbers.

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Are all models of peano arithmetics descibed using first order logic non standard?

It is known that there are non-standard models of Peano Arithmetics when it is described using first order logic. My question is if there is standard model (one which does not contains non-standard ...
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1answer
107 views

Find the representation in Peano system

It's known that all recursive functions are representable in Peano arithmetic. I am trying to find representation of subtraction function $f(x,y)= \left\{\begin{matrix} x-y &if& x>y\\ 0 &...
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0answers
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Ring Theory and Induction

Andrew Boucher has developed a theory called General Arithmetic (GA): http://www.andrewboucher.com/papers/ga.pdf GA is a sub-theory of Peano Arithmetic (PA). If we add an induction schema (IND) to ...
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406 views

Closed under equality

In the Wikipedia article on "Peano axioms" I read this (source): For all $a$ and $b$, if $a$ is a natural number and $a = b$, then $b$ is also a natural number. That is, the natural numbers are ...
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1answer
208 views

How is The strengthened finite Ramsey theorem known to be true for natural numbers?

http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem states that independence of "The strengthened finite Ramsey theorem" from PA is proved by implying consistency of PA. But how do we know ...
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3answers
101 views

showing $\nexists\;\beta\in\mathbb N:\alpha<\beta<\alpha+1$

I want to prove that $\nexists\; \beta\in\mathbb N$ such that $\alpha<\beta<\alpha+1$ for all $\alpha\in\mathbb N$. I just want to use the Peano axioms and $+$ and $\cdot$ If $\alpha<\beta$ ...
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2answers
2k views

Is there a formal definition of “Greater Than”

Intuitively, one can say that $S(n) > n$. But how do we prove it using the Peano Axioms. It seems like I need a formal statement as to what $>$ means.
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1answer
648 views

Is every model of modular arithmetic either even or odd?

Modular Arithmetic (MA) has the same axioms as first order Peano Axioms (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. (http://en.wikipedia.org/wiki/Peano_axioms#First-...
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1answer
688 views

Peano's Postulates Proofs

How can I prove the following two questions: Prove using Peano's Postulates for the Natural Numbers that if a and b are two natural numbers such that a + b = a, then b must be 0? Prove using Peano's ...
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5answers
679 views

Peano postulates

I'm looking for a set containing an element 0 and a successor function s that satisfies the first two Peano postulates (s is injective and 0 is not in its image), but not the third (the one about ...
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2answers
907 views

Induction as Peano Axiom

Let P be some proposition. If we have that $P(0)$ is true and that if $P(n)$ is true, then $P(S(n))$ is true, where $S(n)$ is the successor of natural number $n$. Then we have that $P(n)$ is true for ...
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4answers
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Why is Peano arithmetic undecidable?

I read that Presburger arithmetic is decidable while Peano arithmetic is undecidable, and actually Peano arithmaetic extends Presburger arithmetic just with the addition of the multiplication operator....
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1answer
1k views

Is there a natural model of Peano Arithmetic where Goodstein's theorem fails?

Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It is known to be independent of Peano Arithemtic (PA), and in fact, was the first such purely number theoretic ...

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