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To define natural numbers one can either:

  1. use the Peano axioms in second-order logic;

  2. encode them in set theory as von Neumann ordinals.

The relation between those two definitions of natural numbers is that (2) satisfies (1).

Now, groups are usually defined as sets equipped with operations and axioms. This is akin to (2) above since it is an encoding of groups in set theory.

What would be a definition of groups akin to (1)?

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    $\begingroup$ I don’t have an answer to the question, but it’s worth noting that the natures of groups and the natural numbers are different— the latter is intuitively categorical, while the former is a broad class structures with greatly varying details having a small common set of properties. Very different axiom systems with different purposes. $\endgroup$
    – Ned
    Aug 20, 2020 at 12:58
  • $\begingroup$ Sets are mentioned in the definition of groups just for convenience. The set of axioms can be straightforwardly expressed in first order logic. $\endgroup$
    – egreg
    Aug 20, 2020 at 13:32
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    $\begingroup$ @Ned The nature of groups is not so dissimilar to the one of sets. Still, for sets we have ZF as an axiomatic theory. So it seems reasonable to expect a similar axiomatic theory for groups. $\endgroup$
    – Bob
    Aug 20, 2020 at 16:30
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    $\begingroup$ @Bob It sounds like you want an axiomatization $T$ so that we can reasonably say, the elements of a model of $T$ are what we call "groups" -- but without encompassing a large portion of ZF and just defining groups inside that as in some of the answers below. Maybe category theory offers an approach, but I know absolutely nothing about that. $\endgroup$
    – Ned
    Aug 20, 2020 at 16:53
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    $\begingroup$ ZF describes how the "membership" relation works. And Peano axioms describe "successor". It seems you want axioms that handle groups (rather than elements of a group), but it seems difficult to answer without knowing what operations/relations you would want described. (Homomorphisms between groups?) As @Ned suggests, category theory may help (see Grp), but I wonder if you'll have to all-but rebuild set theory. Like maybe building on ETCS? I'm out of my depth. Maybe replace the [model-theory] tag with [category-theory]? $\endgroup$
    – Mark S.
    Aug 20, 2020 at 18:24

4 Answers 4

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This is too long for a comment. Given the negative reception of the current answers, and the confusion on the original question, I thought I would add what I think is being asked, and why the current answers don't work. (For what it's worth, I did not downvote either answer.)

It appears to me that the OP is trying to compare the class of all groups to the set of natural numbers. In other words, a single group is compared to a single natural number. This is why it is claimed that the definition of groups via axiomatization is akin to (2), rather than (1). Specifically:

  1. Given a model $M$ of ZF, one can define which sets in $M$ are natural numbers using the von Neumann definition: a set is a natural number if and only if blah blah...

  2. Given a model $M$ of ZF, one can define which pairs of sets in $M$ are groups (where the pair consists of the underlying set of the group and the graph of the binary operation) using straightforward axioms: a pair of sets is a group if and only if blah blah...

There are some problems with this, as pointed out in Ned's comment. For example the natural numbers don't change depending on the model of ZF, while on the other hand different models of ZF could disagree about which groups they contain. But even still, the analogy does make sense.

On the other hand, the current answers make a comparison between the set of natural numbers and a particular group (rather than the class of all groups). So this doesn't seem to be what the OP is after in something akin to (1).

In my estimation, something akin to (1) would be an axiomatic theory $T$ which fits the analogy $$ \text{ $T$ : group :: Peano axioms : natural number} $$ So the class of groups would somehow be the "standard model" of $T$ (whatever that might mean) in the same way the natural numbers are the standard model of PA. But already there are issues with how to make this precise. At the very least, I think one would want to represent groups as isomorphism classes or presentations, rather than sets. Then one could make some "meta-axiomatizations" of special sets of group presentations. For example, one could "axiomatize" sets of group presentations that contain the finite cyclic groups and are closed under direct product. Then the finite abelian groups would be something like the "standard model". But this is not nearly rigorous enough.

So I don't know what the right answer is, or if there is a good answer at all. Hopefully what I've said isn't total nonsense, and I haven't misconstrued the original intent of the question.

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  • $\begingroup$ You have perfectly understood my question. $\endgroup$
    – Bob
    Aug 20, 2020 at 14:29
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To axiomatize the collection of all groups is far more difficult than to axiomatize the set of all natural numbers. There are, well, a lot of groups, and in particular it makes little sense, foundationally or practically, to try to axiomatize the mere set of all groups. One approach, as mentioned in comments, is to axiomatize the category of groups. You can read a summary of the results of Leroux on this question in ArnaudD's answer here. The axioms are rather technical, but again, this is perhaps to be expected from such an ambitious question.

A related topic which has been somewhat more thoroughly studied, including a nice exposition by Todd Trimble, is the axiomatization of the category of all sets. This is still complicated, though a bit better; you can get close by saying that the category of sets is a category admitting finite products and equalizers, powersets, such that the terminal object detects equality of morphisms, there is a natural numbers object, and the axiom of choice holds. At least, it's reasonable to say that "the objects of such a category are things we call 'sets' ", though without further axioms such a category may not contain all sets.

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  • $\begingroup$ There are also a lot of sets. But they are axiomatized by ZF which is not so difficult. So it is not obvious to me why it should be difficult for groups. $\endgroup$
    – Bob
    Aug 21, 2020 at 19:14
  • $\begingroup$ You wrote that "it makes little sense, foundationally" but groups were discovered by Galois long before the advent of set theory. It should be interesting to study them on their own without involving set theory, the same way we study natural numbers independently of set theory. $\endgroup$
    – Bob
    Aug 21, 2020 at 19:21
  • $\begingroup$ @Bob I said it makes little sense to study the set or class of all groups, a sentence which implicitly relies on a notion of set theory, but with the analogy to ZF I can follow why you tend to disagree. I think this is really a quite interesting question; I guess it's hard to imagine an axiomatization of the class of all groups that doesn't eventually rely on the existence of certain important functions between the elements of the class (like the group multiplication,) at which point you're in the neighborhood of category theory anyway. It would be great to hear of a way to do this, though! $\endgroup$ Aug 21, 2020 at 20:28
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As essentially listed on the "groups" section of the "List of first-order theories" English Wikipedia page, one approach would be to have a constant symbol $1$ for the identity, a unary function symbol $\cdot^{-1}$ for inverses, and a binary function $\cdot*\cdot$ (we'll write it as an operation), satisfying these axioms:

  • $\forall x,\,1*x=x\land x*1=x$
  • $\forall x,\,x^{-1}*x=1\land x*x^{-1}=1$
  • $\forall x,\,\forall y,\,\forall z,\,(x*y)*z=x*(y*z)$

A group is then a model of these axioms.

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    $\begingroup$ This is the usual definition akin to (2) that I mention in my post. But I am asking here for a definition akin to (1). $\endgroup$
    – Bob
    Aug 20, 2020 at 13:21
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    $\begingroup$ @Bob I disagree. A function symbol is not a function in set theory, it's just a logical symbol. I have made no reference to sets at all. This is superficially similar to a definition of a group in set theory, but different on a logical level. $\endgroup$
    – Mark S.
    Aug 20, 2020 at 13:23
  • $\begingroup$ I disagree, because you wrote that a group is "a model of these axioms". $\endgroup$
    – Bob
    Aug 20, 2020 at 13:27
  • $\begingroup$ @Bob But a group is exactly a model of these axioms. $\endgroup$ Aug 20, 2020 at 15:31
  • $\begingroup$ @AlexKruckman I fully agree. $\endgroup$
    – Bob
    Aug 20, 2020 at 16:05
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I'd argue the definition of groups you are referring to is akin to (1) the Peano axioms and not to (2) the von Neumann construction of $\mathbb N$.

The peano axioms specify abstractly what "a system of natural numbers" is by axiomatizing properties:

A set $N$ together with a map $s\colon N\to N$ is called a system of natural numbers if the axioms … are satisfied.

That is exactly what the usual definition of groups does:

A set $G$ together with a map $m\colon G\times G\to G$ is called a group if the axioms … are satisfied.

The important difference is that in the case of systems of natural numbers, all models satisfying the axioms turn out to be isomorphic. So there is only one concept of "natural numbers" up to isomorphism and one model would be the construction of von Neumann. In the case of groups, there are many non-isomorphic models satisfying the axioms.

The analogy of the von Neumann construction would be any explicitly constructed group by giving a concrete set and writing down how to multiply its elements.

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    $\begingroup$ Would the down voters care to explain what is wrong with this answer? $\endgroup$
    – Christoph
    Aug 20, 2020 at 13:33
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    $\begingroup$ I am one of them. The reason is that I think that you misunderstood my question. This might be because I was not clear enough in asking. See the answer of @halrankard that reformulates my question more clearly. $\endgroup$
    – Bob
    Aug 20, 2020 at 14:34
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    $\begingroup$ I see, you are comparing $\mathbb N$ and the class $\mathrm{Grp}$ of all groups. Please clarify that in your question instead of just down voting answers that don't fit your unspoken needs. $\endgroup$
    – Christoph
    Aug 24, 2020 at 14:44
  • $\begingroup$ Thanks to the proposed answers and comments, I now realize that my question was not clear. I did not realize that at the time of asking. Please do not hesitate to edit my question. In the meantime, I will cancel my unfair downvotings. $\endgroup$
    – Bob
    Aug 25, 2020 at 9:24

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