Logical Explanation
By Godel's incompleteness theorem one cannot prove the consistency of ZFC using only the axioms of ZFC, and consequently one cannot prove the consistency of ZFC+H for any hypothesis H using only ZFC+H. For this reason the aim of a consistency proof is to prove the consistency of ZFC + H relative to consistency of ZFC. Such problems are known as problems of relative consistency. In fact one proves
(*)
We will give the general schema of relative consistency proofs. Because any proof is finite it uses finite number of axioms.
For any given proof ZFC can verify validity of this proof. This is provable by induction by the length of the proof.
Now we obtain
If we prove following
(**)
we can conclude that
which is equivalent to
which gives (*). Core of relative consistency proof is proving of (**). One have to construct ZFC proof of Con(T + H) for any given finite set T of ZFC axioms (by ZFC instruments of course). (No universal proof of Con(T + H) of course.)
In ZFC is provable that for any condition p the set of formulas (evaluated by names) forced by p is deductive closed. Also, for any ZFC axiom ZFC proves that this axiom is forced by 1. Then proving that there is at least one condition which forces H suffices.
In case of Boolean valued forcing procedure is similar – one have to prove that Boolean value of H is not 0.
Another approach is using reflection theorem. For any given finite set of ZFC axioms there is ZFC proof that this set of axioms has countable transitive model. For any given finite set T of ZFC axioms there is finite set T' of ZFC axioms such that ZFC proves that if countable transitive model M satisfies T' then M satisfies T. One have to prove that there is finite set T" of ZFC axioms such that if countable transitive model M satisfies T" then M satisfies considered hypothesis H. Then for any given finite set T of ZFC axioms ZFC proves Con(T + H).
Sometimes in (**) some stronger theory S than ZFC is used for proving Con(T + H). Then we have proof of consistency of ZFC + H relative to consistency of S. Note that, where ZFL is ZF + V = L (axiom of constructibility).
Read more about this topic: Forcing (mathematics)
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