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I have two definitions from Wikipedia:

  1. The usual set-theoretic definition of a group at: http://en.wikipedia.org/wiki/Glossary_of_group_theory

  2. The first-order definition of a group at: http://en.wikipedia.org/wiki/List_of_first-order_theories#Groups

No domain of quantification is given in 2. Was this deliberate? Are the quantifiers there, as some have suggested to me, truly unbounded? Are they really talking there about an operator defined on every real and imagined object in the universe? Is first order group theory a separate field of study from set-theoretic group theory? If so, are there any important results in first order group theory than cannot be obtained in set-theoretic group theory.

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In a first-order theory, all quantifiers implicitly range over all objects in the theory's universe -- that is, once a group that is a model of the theory is given, they quantify over all elements of that group. –  Henning Makholm Jan 4 '12 at 16:50
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@Dan: To reinforce Henning's point: The universe of discourse of a first order theory is not necessarily a universe of sets. Rather, it is just a model of that theory, so for the theory of groups a universe is just a group. Set theory is a bit unusual in that a model of set theory is not a set but rather a universe of sets... –  Zhen Lin Jan 4 '12 at 17:02
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@Zhen I upvoted your comment, but then I thought that Set theory only "appears" to be different from other theories. In reality its universe of discourse is just informally called "universe" and its objects are informally called "sets". So it is a "universe of sets". But have these "sets" the same properties that other universes of discourse have in other first order theories? –  magma Jan 7 '12 at 17:39

2 Answers 2

up vote 6 down vote accepted

The language is not necessarily the one described in the article. More common is to have a single binary function symbol.

That said, the symbols of the language are precisely that, symbols. In particular, $\forall$ and $\exists$ are symbols, they do not range over anything.

After we have described the language, and the axioms, which are certain strings of symbols, we obtain a theory $T$. We are interested in models of that theory. Suppose for simplicity that our language has a single binary function symbol $p$. A model $\mathcal{A}$ of our theory is a non-empty set $A$, together with an honest to goodness binary function $p_\mathcal{A}\colon A\times A \to A$ such that under the usual definition of truth, all the axioms of $T$ are true in the structure with underlying set $A$ and binary function $p_\mathcal{A}$.

The details of the definition of truth are a bit too lengthy to give here. Roughly speaking, we define truth in $\mathcal{A}$ of sentences $\varphi$ of our language by an induction on the complexity of $\varphi$. As a simple example, we say that $\varphi\land \psi$ is true in $\mathcal{A}$ if both $\varphi$ and $\psi$ are true in $\mathcal{A}$. Note that $\land$ is a formal symbol. This part of the definition of truth in a sense assigns meaning to the formal symbol $\land$. But in principle the syntax, which deals with uninterpreted formal symbols, is kept strictly separate from the semantics (models). That separation is not always strictly maintained, because the most interesting questions deal with the interaction between syntax and semantics. The link between the two is the definition of truth in a structure.

To partly answer your question, we look at the sentence
$$\exists x\forall y (p(x,y)=y).$$ The above axiom is intended to assert the existence of a left-identity, but it is just a string of symbols. For this axiom to be true in $\mathcal{A}$ means that there is an element $e \in A$ such that for all $a\in A$, $f_A(e,a)=a$.

Thus it is at the model stage that the formal symbols $\forall$ and $\exists$ are interpreted as working like quantifiers in the usual informal sense. The technical details of the definition of truth in a structure $\mathcal{A}$ take care of that. Since we are working in a specific model, the quantification is always over a completely specific set $A$, so the universe is always a specific set.

To sum up, the models of the first-order Theory of Groups are precisely the set-theoretic groups.

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I'm not sure I understand the need for a "first-order" group theory. Are there any theorems about groups in general that can be obtained using the first-order definition (2), but not by using the set-theoretic definition (1)? –  Dan Christensen Jan 4 '12 at 17:52
    
There is a quite large literature on first-order group theory. One moderately interesting result, for example, is that there is an algorithm which will tell ou, given a first-order sentence $\varphi$, whether that sentence is rue in all Abelian groups. There are many results of a similar character for other algebraic structures, such as fields. Any sentence $\varphi$ of the first-order theory of fields of characteristic $0$ is true in an algebraically closed field $F$ iff it is true in the complex numbers. There are more striking deeper results, such as the Ax-Kochen Theorem. –  André Nicolas Jan 4 '12 at 17:59
    
Is this result not obtainable using the set-theoretic definition and the usual rules and axioms of logic and set theory? –  Dan Christensen Jan 4 '12 at 18:09
    
Is there a theorem stated in the language of set-theoretic group theory that can only be obtained using first-order group theory. –  Dan Christensen Jan 4 '12 at 18:41
    
@Dan Christensen: Almost certainly not. But there are a number of instances of first proofs being obtained using model-theoretic techniques. –  André Nicolas Jan 4 '12 at 18:51

Your observation is very good. Group theory can be stated outside of any specific Set Theory (there are many, as one can see in the second of your referenced Wikipedia pages). Same thing could be said for Ring Theory or other algebraic theories. In particular Category Theory is a first order theory (not - yet - listed in that Wikipedia page) which is so general that some propose to replace set theory (or theories) with some category theory-related axioms. Mac Lane's "Categories for the Working mathematicians" explaisn the situation very clearly (Chap. 1 Sec. 1):

First we describe categories by means of axioms, without using any set theory, and calling them "metacategories".

Then in Chap 1, Sec 2:

A category (as distinguished from a metacategory) will mean any interpretation of the category axioms within set theory

Likewise one could say (in one's mind at least, since this is not done in practice AFAIK): first order axioms describe metagroups, while interpretations of these axioms within a specific set theory describe groups. One can get a better understanding of these points of view by reading an introduction to model theory for example starting with Wikipedia: http://en.wikipedia.org/wiki/Model_theory

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