Questions tagged [peano-axioms]

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

564 questions
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Why does induction have to be an axiom?

I noticed that there is an axiom that says that if $S(n)\implies S(n+1)$, and $S(1)$ is true, then $\forall n \in \Bbb N, S(n).$ My question is why is this an axiom? why can't we derive this from the ...
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Proof that an injective system with $i\notin\textrm{ran}{f}$ has a Peano subsystem.

I follow this link, in particular Exercise 2 at the bottom of page 3. Def 1. A System is a tripple $(X,i,f)$, where $X$ is a set, $i$ is called initial element, and $f$ is a function $X\to X$. Def 2....
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Mathematical Induction and Peano Arithmetic

Peano Arithmetic cannot employ Induction for any ε0 ordering. My question is too easy to be interesting and there is a reason obviously for why it has a negative answer. Can you please provide it for ...
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How can we be so sure that we don't live in Pudlak's inconsistent world?

In his Logical Foundation of Mathematics and Computational Complexity (2013), Pavel Pudlak invites the readers to ponder about fictitious people whose natural numbers are nonstandard. His exposition ...
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Can Peano arithmetic prove 0=1 in a standard number of steps (in a non-standard model)?

Let $M$ be a model of $PA + \lnot Con(PA)$. Therefore, there exists an object $p \in \mathbb N_M$ encoding the the $PA \vdash 0=1$. Is there such a model $M$ and $p \in \mathbb N_M$ such that the ...
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Is $K_{n-1}$ a $\Sigma_{n-1}$-elementary substruture of $K_n$ [closed]
Is $K_{n-1}$ a $\Sigma_{n-1}$ elementary substruture of $K_n$? Let $M$ be any non-standard model of PA. $K_n$ is define to be the set of $\Sigma_n$-definable elements of $M$. I have a feeling the ...