FOM: RE: Axion's objetive

[Karl Cooper]

>    1)there are not infinitely many elements
>    2)for the naturals we can get the 0,the sucesor axiom but not the 
> induction one.
>    3)all the infinite sets only can be given as the rule ( finite ) to get
> 
> its elements..
> 
>  Does someboddy know some work near this?
>    
Well, you should take a look at Harvey Friedman's posting relating
subsystems of arithmetic to subsystems of set theory ("76: Finite set
theories"):
http://www.math.psu.edu/simpson/fom/postings/9912/msg00042.html
Others on the list probably could tell you the state of discussion among
set theorists of such subsystems of set theory.
>