There is those one Q5 to Q7 in <https://en.wikipedia.org/wiki/Hilbert_system#Formal_deductions>


But I know the axioms of Boolean algebra were simplified to this <https://en.wikipedia.org/wiki/Wolfram_axiom>


I was wondering if similar researchs have been done on quantifiers?



Is there a simpler axiomatization for the quantifiers?
There is [The Epsilon Calculus](http://plato.stanford.edu/entries/epsilon-calculus/) developed by David Hilbert during the 20s.


It is based on the $ε$ *symbol* :



> 
> if $A$ is a formula and $x$ is a variable, $εx \ A$ is a *term*
> 
> 
> 


with the axiom (Hilbert's “transfinite axiom”) :



> 
> $A(x) → A(εx A)$.
> 
> 
> The intended interpretation is that $εx \ A$ denotes some $x$ satisfying $A$, if there is one. 
> 
> 
> 


Quantifiers can be defined as follows:



> 
> 
> > 
> > $∃x A(x) ≡ A(εx A)$
> > 
> > 
> > $∀x A(x) ≡ A(εx (¬A)).$
> > 
> > 
> > 
> 
> 
> 




---


See the use of epsilon notation in [Nicolas Bourbaki](https://en.wikipedia.org/wiki/Nicolas_Bourbaki), [Elements of Mathematics : Theory of sets](https://books.google.it/books?id=7eclBQAAQBAJ&pg=PA36) (1968 - French ed. 1958), page 20 and page 36 [with $\tau\_x$ in place of $εx$].


