[FOM] Questions about a restricted ZF

[Paul Budnik]

I am attempting to formalize my concept of objective mathematics. It 
appears that I need a restricted ZF without the power set axiom and with 
quantifiers in the axiom of replacement limited to the integers plus 
universal (no existential) quantifiers over subsets of the integers. 
(Being a subset of the integers must be defined as a property and not by 
a set.) 

What I think this defines are countable admissible ordinals less than 
some limit and the sets constructible from them. Is this correct? Is 
there research on the hierarchy that results from these restrictions on 
quantifiers?

I am not suggesting an alternative to ZF.  I think all the objective 
statements decidable in ZF are correctly decided. The purpose is to 
separate out statements that have a definite objective truth value from 
those than I think only have relative meaning like the continuum hypothesis.

Paul Budnik
www.mtnmath.com