Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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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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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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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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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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Why do we take the axiom of induction for natural numbers (Peano arithmetic)?
More precisely, when we define the set of natural numbers $\mathbb{N}$ using the Peano axioms, we assume the following:
$0\in\mathbb{N}$
$\forall n\in\mathbb{N} (S(n)\in\mathbb{N})$
$\forall n\in\...
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Why is Peano arithmetic undecidable?
I read that Presburger arithmetic is decidable while Peano arithmetic is undecidable. Peano arithmetic extends Presburger arithmetic just with the addition of the multiplication operator. Can someone ...
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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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