[FOM] ZFC-proofs of arithmetical statements
joeshipman@aol.com
joeshipman at aol.com
Tue Apr 14 02:00:39 EDT 2009
Consider the following classes of ZFC-theorems about the integers (in a
version of ZFC with both Separation and Replacement):
A) Arithmetical Theorems which have a proof not involving the Axiom of
Infinity [equivalent to PA-theorems]
B) Arithmetical Theorems which have a proof not involving the Power Set
Axiom
C) Arithmetical Theorems which have a proof not involving the
Replacement Axiom-Scheme
D) All Arithmetical Theorems
Is it correct to say that each class strictly contains the previous
one? Or do I need to replace B), for example, with
B') Arithmetical Theorems which have a proof not involving both the
Power Set Axiom and the Axiom of Infinity?
What other interesting classes of arithmetical theorems can be defined
in terms of which Axiom(-Schemes) of ZFC are needed? (Choice and
Regularity and Extensionality ought not to matter in the presence of
all the others, but perhaps one of them might be necessary if others
are absent.)
-- JS
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