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Definition of an axiom

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https://www.novabbs.com/tech/article-flat.php?id=86273&group=sci.math#86273

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Subject: Definition of an axiom
From: pepste...@gmail.com (Paul Epstein)
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by: Paul Epstein - Wed, 22 Dec 2021 17:09 UTC

Would the folling induction schema on the natural numbers be considered
an axiom? (ϕ(x; 0) ∧ ∀x(ϕ → ϕ(x; Sx))) → ∀xϕ
I think not, but I'm not completely sure.
Is the reason it's not an axiom that it quantifies over all propositions ϕ, whereas an "axiom"
can only quantify over all elements of an underlying set.
Or is it actually an axiom?
And, if it is an axiom, why is Modus Ponens (the trick which entitles you
to assert B if you've proved both A and (A implies B) not considered an
axiom?

Thank You,

Paul Epstein

Re: Definition of an axiom

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Subject: Re: Definition of an axiom
From: fredjeff...@gmail.com (FredJeffries)
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by: FredJeffries - Wed, 22 Dec 2021 20:57 UTC

On Wednesday, December 22, 2021 at 9:09:34 AM UTC-8, Paul Epstein wrote:
> Would the folling induction schema on the natural numbers be considered
> an axiom? (ϕ(x; 0) ∧ ∀x(ϕ → ϕ(x; Sx))) → ∀xϕ
> I think not, but I'm not completely sure.
> Is the reason it's not an axiom that it quantifies over all propositions ϕ, whereas an "axiom"
> can only quantify over all elements of an underlying set.
> Or is it actually an axiom?
> And, if it is an axiom, why is Modus Ponens (the trick which entitles you
> to assert B if you've proved both A and (A implies B) not considered an
> axiom?

https://en.wikipedia.org/wiki/Axiom_schema
https://mathworld.wolfram.com/AxiomSchema.html

Subject	Author
Definition of an axiom	Paul Epstein
Re: Definition of an axiom	FredJeffries