# Questions tagged [peano-axioms]

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

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### Does transfinite induction follow from second-order induction?

Second-order induction is defined in the Peano Axioms as follows: for all formulas $φ$, $φ(0) \rightarrow (\forall v(A(v) \rightarrow φ(s(v))) \rightarrow \forall n φ(n))$, where $s$ is some ...
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### Using the Compactness Theorem for First-order Logic to find a model

The problem I am trying to understand is as follows, which was posed as an example in one of my lectures. The motivation of the problem is to apply the compactness theorem: Consider the language with ...
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### Peano arithmetics formulae examples [closed]

Help me please to come up with an example of two arithmetic formulae $\varphi$ and $\psi$ such that $PA\vdash\varphi\vee\psi$, but neither $\varphi$ nor $\psi$ is derivable in $PA$ ($PA$ is Peano ...
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### Can PA be interpreted in a theory about a total function on sets that is sensitive to proper subsethood?

Language tri-sorted FOL with identity, with lower cases representing individuals, and upper cases representing sets of them, and asterisked upper cases for sets of sets of individuals.A one place ...
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### Prove that $+:\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is a binary operation

The question is in the title. The definition of $+:\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is given as follows: $$\forall m,n \in \mathbb{N}: [n+1 := n'] \land [n+m' := (n+m)']$$ where $n'$ ...
I'm studying for an exam of mathematical logic. This question envolves the Peano axioms, I think. Prove that, for all $n \in \omega$, $n \notin n$. It's kind of obvious that it's true but I don't ...