[FOM] Cantor Theorem

Ross A. Finlayson apex at apexinternetsoftware.com
Tue Feb 11 02:25:17 EST 2003

There are anti-foundational set theories.

I have a problem with "models" vis-a-vis theories and "classes" 
vis-a-vis sets.  I think a model is a type of theory and a class a type 
of set.

ZF is exactly what it describes:  a set theory where all sets are 
regular.  Where an axiom of ZF is decided by other axioms of ZF then it 
is a theorem of a reduced ZF.  That is also to say that a theory with 
axioms inconsistent with or negatory against "axioms" of ZF would 
possibly contradict other axioms, among their independence results.

Sometimes I use these words cavalierly, I do try to use them correctly.

I think some or all axioms of ZF are theories of a reduced theory.  
Also, I think that a reduced theory may have no axioms yet may have 
decidable theorems.  My example is a reduced theory with identity as a 
tautology and excluded middle.

I think Choice is a theorem and that V=L is decided by ZF.

I think that the Cantorian powerset mapping result, a proof, is 
restrictive and that to be ignored that it requires qualifications.  I 
don't see the Cantorian antidiagonalization about reals and naturals as 
a proof.

About the powerset result, I rationalize a powerset with the element of 
null or N as the order type of set as the element mapped or not mapped, 
in conjunction with bridging results mapping other infinite sets, to be 
the rationalization of the Cantorian result mapping infinite to 
transfinite sets.  As well, a set of all sets is its own powerset.

Ross Finlayson

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