First of all, induction only really works on well-ordered, countable sets: things like $\mathbb{Z}$ and $\mathbb{N}$, where you can always define a "next" number. You can well-order $\mathbb{Q}$, but not in any way that makes induction useful: the most common well-ordering of $\mathbb{Q}$ starts $0,1,-1,2,\frac12,-\frac12,-2,3,\frac13,-\frac13,-3,4,\frac32,\frac23,\frac14,-\frac14,-\frac23,-\frac32,-4$ and continues on like that. (See if you can spot the pattern!) For any rational number, it really doesn't make sense to talk about the "next" rational number, and so induction isn't really useful.

$\mathbb{R}$ has all the issues of $\mathbb{Q}$, but it's not even countable. An induction on $\mathbb{R}$ that considers each real number separately requires all sorts of weirdness like the Axiom of Choice and maybe even some form of Transfinite induction in order to make it work, and it'll be a mess. Of course, there are ways around this, like slicing up \mathbb{R} into countably many pieces, but that's usually just standard induction with intervals.

Incidentally, though, we do need to establish that multiplication on $\mathbb{R}$ is commutative, and the way we do it depends on the way we're constructing $\mathbb{R}$. Sometimes, we poof $\mathbb{R}$ into existence using the magic spell, "the unique Dedekind-complete ordered field up to isomorphism" and then we don't need to do much work; the field axioms establish that the operation we call "multiplication" must be associative. Careful, though; if you're doing this for the first time, you may want to establish that this approach actually does give a unique, well-defined set, and that this set is indeed the "real numbers" that you've grown to know.

The other way to define real numbers is by "building them up" in some way from the rationals (which, incidentally, are defined by "building up" from the integers, which are in turn defined by "building up" from the naturals). There are a few ways of doing this, and some of them make showing commutativity easier than others do, but either way we do have to prove it. It just turns out that induction is usually totally useless, because by the time you've shown that this new set behaves the way you want it to behave (i.e. being Dedekind-complete, closed under certain operations, etc) you've already probably shown that the multiplication operator is commutative. If you're curious about how that's done, here's a paper that goes into depth on two of the most common ways to construct the reals, Dedekind cuts and Cauchy sequences.

The naturals, on the other hand, are far more fundamental; there's really not much that you can "build up" from in a rigorous and meaningful way, and there's really only one commonly accepted definition of the naturals as far as I know. In that definition, commutativity is a theorem, not an axiom. I suppose if you really tried, you could find some contrived definition of the naturals where commutativity of multiplication has to be an axiom, but it's so much simpler to just use Peano and induct.

First of all, induction only really works on well-ordered, countable sets: things like $\mathbb{Z}$ and $\mathbb{N}$, where you can always define a "next" number. You can well-order $\mathbb{Q}$, but not in any way that makes induction useful: the most common well-ordering of $\mathbb{Q}$ starts $0,1,-1,2,\frac12,-\frac12,-2,3,\frac13,-\frac13,-3,4,\frac32,\frac23,\frac14,-\frac14,-\frac23,-\frac32,-4$ and continues on like that. (See if you can spot the pattern!) For any rational number, it really doesn't make sense to talk about the "next" rational number, and so induction isn't really useful.

$\mathbb{R}$ has all the issues of $\mathbb{Q}$, but it's not even countable. An induction on $\mathbb{R}$ that considers each real number separately requires all sorts of weirdness like the Axiom of Choice and maybe even some form of Transfinite induction in order to make it work, and it'll be a mess. Of course, there are ways around this, like slicing up $\mathbb{R}$ into countably many pieces, but that's usually just standard induction with intervals.

Incidentally, though, we do need to establish that multiplication on $\mathbb{R}$ is commutative, and the way we do it depends on the way we're constructing $\mathbb{R}$. Sometimes, we poof $\mathbb{R}$ into existence using the magic spell, "the unique Dedekind-complete ordered field up to isomorphism" and then we don't need to do much work; the field axioms establish that the operation we call "multiplication" must be associative. Careful, though; if you're doing this for the first time, you may want to establish that this approach actually does give a unique, well-defined set, and that this set is indeed the "real numbers" that you've grown to know.

The other way to define real numbers is by "building them up" in some way from the rationals (which, incidentally, are defined by "building up" from the integers, which are in turn defined by "building up" from the naturals). There are a few ways of doing this, and some of them make showing commutativity easier than others do, but either way we do have to prove it. It just turns out that induction is usually totally useless, because by the time you've shown that this new set behaves the way you want it to behave (i.e. being Dedekind-complete, closed under certain operations, etc) you've already probably shown that the multiplication operator is commutative. If you're curious about how that's done, here's a paper that goes into depth on two of the most common ways to construct the reals, Dedekind cuts and Cauchy sequences.

The naturals, on the other hand, are far more fundamental; there's really not much that you can "build up" from in a rigorous and meaningful way, and there's really only one commonly accepted definition of the naturals as far as I know. In that definition, commutativity is a theorem, not an axiom. I suppose if you really tried, you could find some contrived definition of the naturals where commutativity of multiplication has to be an axiom, but it's so much simpler to just use Peano and induct.

First of all, induction only really works on well-ordered, countable sets: things like $\mathbb{Z}$ and $\mathbb{N}$, where you can always define a "next" number. You can well-order $\mathbb{Q}$, but not in any way that makes induction useful: the most common well-ordering of $\mathbb{Q}$ starts $0,1,-1,2,\frac12,-\frac12,-2,3,\frac13,-\frac13,-3,4,\frac32,\frac23,\frac14,-\frac14,-\frac23,-\frac32,-4$ and continues on like that. (See if you can spot the pattern!) For any rational number, it really doesn't make sense to talk about the "next" rational number, and so induction isn't really useful.

$\mathbb{R}$ has all the issues of $\mathbb{Q}$, but it's not even countable. An induction on $\mathbb{R}$ that considers each real number separately requires all sorts of weirdness like the Axiom of Choice and maybe even some form of Transfinite induction in order to make it work, and it'll be a mess. Of course, there are ways around this, like slicing up \mathbb{R} into countably many pieces, but that's usually just standard induction with intervals.

Incidentally, though, we do need to establish that multiplication on $\mathbb{R}$ is commutative, and the way we do it depends on the way we're constructing $\mathbb{R}$. Sometimes, we poof $\mathbb{R}$ into existence using the magic spell, "the unique Dedekind-complete ordered field up to isomorphism" and then we don't need to do much work; the field axioms establish that the operation we call "multiplication" must be associative. Careful, though; if you're doing this for the first time, you may want to establish that this approach actually does give a unique, well-defined set, and that this set is indeed the "real numbers" that you've grown to know.

The other way to define real numbers is by "building them up" in some way from the rationals (which, incidentally, are defined by "building up" from the integers, which are in turn defined by "building up" from the naturals). There are a few ways of doing this, and some of them make showing commutativity easier than others do, but either way we do have to prove it. It just turns out that induction is usually totally useless, because by the time you've shown that this new set behaves the way you want it to behave (i.e. being Dedekind-complete, closed under certain operations, etc) you've already probably shown that the multiplication operator is commutative. If you're curious about how that's done, here's a paper that goes into depth on two of the most common ways to construct the reals, Dedekind cuts and Cauchy sequences.

The naturals, on the other hand, are far more fundamental; there's really not much that you can "build up" from in a rigorous and meaningful way, and there's really only one commonly accepted definition of the naturals as far as I know. In that definition, commutativity is a theorem, not an axiom. I suppose if you really tried, you could find some contrived definition of the naturals where commutativity of multiplication has to be an axiom, but it's so much simpler to just use Peano and induct.

First of all, induction only really works on well-ordered, countable sets: things like $\mathbb{Z}$ and $\mathbb{N}$, where you can always define a "next" number. You can well-order $\mathbb{Q}$, but not in any way that makes induction useful: the most common well-ordering of $\mathbb{Q}$ starts $0,1,-1,2,\frac12,-\frac12,-2,3,\frac13,-\frac13,-3,4,\frac32,\frac23,\frac14,-\frac14,-\frac23,-\frac32,-4$ and continues on like that. (See if you can spot the pattern!) For any rational number, it really doesn't make sense to talk about the "next" rational number, and so induction isn't really useful.

$\mathbb{R}$ has all the issues of $\mathbb{Q}$, but it's not even countable. An induction on $\mathbb{R}$ that considers each real number separately requires all sorts of weirdness like the Axiom of Choice and maybe even some form of Transfinite induction in order to make it work, and it'll be a mess. Of course, there are ways around this, like slicing up \mathbb{R} into countably many pieces, but that's usually just standard induction with intervals.

Incidentally, though, we do need to establish that multiplication on $\mathbb{R}$ is commutative, and the way we do it depends on the way we're constructing $\mathbb{R}$. Sometimes, we poof $\mathbb{R}$ into existence using the magic spell, "the unique Dedekind-complete ordered field up to isomorphism" and then we don't need to do much work; the field axioms establish that the operation we call "multiplication" must be associative. Careful, though; if you're doing this for the first time, you may want to establish that this approach actually does give a unique, well-defined set, and that this set is indeed the "real numbers" that you've grown to know.

The other way to define real numbers is by "building them up" in some way from the rationals (which, incidentally, are defined by "building up" from the integers, which are in turn defined by "building up" from the naturals). There are a few ways of doing this, and some of them make showing commutativity easier than others do, but either way we do have to prove it. It just turns out that induction is usually totally useless, because by the time you've shown that this new set behaves the way you want it to behave (i.e. being Dedekind-complete, closed under certain operations, etc) you've already probably shown that the multiplication operator is commutative. If you're curious about how that's done, here's a paper that goes into depth on two of the most common ways to construct the reals, Dedekind cuts and Cauchy sequences.

The naturals, on the other hand, are far more fundamental; there's really not much that you can "build up" from in a rigorous and meaningful way, and there's really only one commonly accepted definition of the naturals as far as I know. In that definition, commutativity is a theorem, not an axiom. I suppose if you really tried, you could find some contrived definition of the naturals where commutativity of multiplication has to be an axiom, but it's so much simpler to just use Peano and induct.