TITLE: getting rid of existential quantifiers
QUESTION [14 upvotes]: It seems to me that for most of the twentieth century, axiomatic foundations for mathematical theories were constructed with the (mostly allied) goals of minimizing the number of primitive notions and minimizing the number of axioms.  But one could equally well be guided by the goal of minimizing the logical depth of the axioms, i.e., minimizing the use of quantifiers.
Consider group theory, for instance.  I have seen formalizations of group theory that say "There exists an element $h \in G$ such that for all $g \in G$, $gh=hg=g$" and then go on to prove that $h$ is unique and to name it $e$.  But I have also seen formalizations that say "A group is a set $G$ equipped with an element $e$ such that..." which eliminates the need for a quantifier in an axiom.  Likewise, if one includes the operation $g \mapsto g^{-1}$ as a primitive, one can avoid the axiom "For all $g$ there exists $h$ such that $gh=hg=e$" (or worse, "For all $g$ there exists $h$ such that for all $h'$, $ ghh'=hgh'=h'gh=h'hg"$, which I haven't seen, but which I can imagine a certain sorts of purist preferring!).
I know that in constructive mathematics, the status of existential quantifiers is suspect to begin with, so I imagine that there's already quite a bit of writing in foundations and philosophy of mathematics (and maybe computable mathematics as well) that addresses this issue.  Some pointers to relevant literature would be appreciated.

REPLY [2 votes]: Hilbert's epsilon-calculus  is  another method to completely get rid of all existential quantifiers as well as all universal quantifiers.  It is almost, but not quite, entirely unlike Skolemization.  
For any formula $A(x)$, the term $\epsilon x(A(x))$ is read as "any $x$ satisfying property $A$, if there is any -- and anything else otherwise".  Thus the formula $A( \,\epsilon x(A(x))$ says that $A$ holds for such an $x$, i.e., $\exists x\, A(x)$ holds.  The formula $\lnot A( \ \epsilon x(A(x) \ )$  then says that $A$ does not even hold for the $x$ which, if at all possible, satisfies $A$, or in other words:  $\lnot \exists x(A(x))$.   
The universal formula $\forall x (B(x))$ can therefore be rewritten as 
$B(\ \epsilon x (\lnot B(x))\ )$. ("Pick some $x$, any $x$ -- if there is one at all -- which fails property $B$.  Then $B$ will hold even of this $x$. So you failed to pick a counterexample. So $B$ holds for all $x$.") 
Theoretically this is wonderful -- not need for quantifiers, and you can use $\epsilon$ like a real term, you can plug terms into terms, etc.   In practice it did not catch on at all -- probably because formulas get very complicated very quickly, and also because the "meaning" of $\forall x (B(x))$ is so much easier to understand than the meaning of $B(\ \epsilon x (\lnot B(x))\ )$.