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Heavily changed the post to avoid false insinuations and severe mischaracterizations.
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MaximusIdeal
MaximusIdeal
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I figured this question is better suited to this stackexchange. I give some mathematical details, but this is primarily an HSM question.

According to this post, the existence of multiplication in Peano arithmetic is an axiom rather than a definition. This is surprising toleads me, because all sources I have looked at so far presented to the following questions below.

How does this square with the fact that Peano axioms are usually presented as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication.?

As the post says, Peano axioms are usually introduced in the context of ZF(C) set theory, in which the recursion theorem can be proven from ZF axioms, so this partially explains why sources provideWas multiplication by a definition (which requires the recursion theorem) rather thanpostulated as an axiomatic postulation.

My own interestaxiom in this question comes from my attempt to derive the usual properties of natural numbers using the usual five Peano axioms. I noticed that to define addition and multiplicationPeano's original work, you need the recursion theorem. This link shows thator was it is possible to prove the recursion theorem, but that requires an order relation (the link takes the ordering of natural numbers for granted), and to define an ordering you need again some form of recursion. (I'm aware an ordering can be defined in the context of von Neumann ordinals, but that's not the context I'm talking about.) No matter what I've tried, I couldn't makegiven by a non-circular derivation.definition?

So I want to confirm. Historically, was multiplication postulated as an axiomIs the person in Peano axioms? Are the modern sources actually misrepresenting Peano axiomspost wrong?

I figured this question is better suited to this stackexchange. I give some mathematical details, but this is primarily an HSM question.

According to this post, the existence of multiplication in Peano arithmetic is an axiom rather than a definition. This is surprising to me, because all sources I have looked at so far presented Peano axioms as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication.

As the post says, Peano axioms are usually introduced in the context of ZF(C) set theory, in which the recursion theorem can be proven from ZF axioms, so this partially explains why sources provide multiplication by a definition (which requires the recursion theorem) rather than an axiomatic postulation.

My own interest in this question comes from my attempt to derive the usual properties of natural numbers using the usual five Peano axioms. I noticed that to define addition and multiplication, you need the recursion theorem. This link shows that it is possible to prove the recursion theorem, but that requires an order relation (the link takes the ordering of natural numbers for granted), and to define an ordering you need again some form of recursion. (I'm aware an ordering can be defined in the context of von Neumann ordinals, but that's not the context I'm talking about.) No matter what I've tried, I couldn't make a non-circular derivation.

So I want to confirm. Historically, was multiplication postulated as an axiom in Peano axioms? Are the modern sources actually misrepresenting Peano axioms?

I figured this question is better suited to this stackexchange. I give some mathematical details, but this is primarily an HSM question.

According to this post, the existence of multiplication in Peano arithmetic is an axiom rather than a definition. This leads me to the following questions below.

How does this square with the fact that Peano axioms are usually presented as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication?

Was multiplication postulated as an axiom in Peano's original work, or was it given by a definition?

Is the person in the post wrong?

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MaximusIdeal
MaximusIdeal
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  • 11
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MaximusIdeal
MaximusIdeal
  • 403
  • 2
  • 11

Is multiplication postulated axiomatically in Peano arithmetic?

I figured this question is better suited to this stackexchange. I give some mathematical details, but this is primarily an HSM question.

According to this post, the existence of multiplication in Peano arithmetic is an axiom rather than a definition. This is surprising to me, because all sources I have looked at so far presented Peano axioms as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication.

As the post says, Peano axioms are usually introduced in the context of ZF(C) set theory, in which the recursion theorem can be proven from ZF axioms, so this partially explains why sources provide multiplication by a definition (which requires the recursion theorem) rather than an axiomatic postulation.

My own interest in this question comes from my attempt to derive the usual properties of natural numbers using the usual five Peano axioms. I noticed that to define addition and multiplication, you need the recursion theorem. This link shows that it is possible to prove the recursion theorem, but that requires an order relation (the link takes the ordering of natural numbers for granted), and to define an ordering you need again some form of recursion. (I'm aware an ordering can be defined in the context of von Neumann ordinals, but that's not the context I'm talking about.) No matter what I've tried, I couldn't make a non-circular derivation.

So I want to confirm. Historically, was multiplication postulated as an axiom in Peano axioms? Are the modern sources actually misrepresenting Peano axioms?