Re: Confusion about gödels proof for con(ZF+CH)

On 2008-06-07, in sci.logic, Bjarne wrote:
My problem is the first part. How did he know that consistency
implied the existence of a model? I thought that this was not proven
until 1947? I have seen some people refer to this result as "Gödel's
completeness theorem", but I believe that this is not accurate?

The completeness theorem, that a theory is consistent iff it has a
model, was proved in Godel's 1929 dissertation and published in 1930.

Also, isn't V more that just a model?

As the term is usually defined, V is not a model at all, but rather
the collection of all sets.

There is no need to consider any models at all. In the proof Godel
established that there is a formula L(x) such that if we replace every
quantifier Ax(...) with Ax(L(x) --> ...) and Ex(...) with Ex(L(x) &
....) -- in technicalese, relativising to L -- the axioms of ZF remain
provable (in ZF), and further, the generalised continuum hypothesis
and the axiom of choice become provable.

The formula L(x) says that x belongs to the constructible universe L,
obtained by transfinite recursion as

L_0 = 0
L_alpha + 1 = L_alpha union Def(L_alpha)
L_lambda = union of L_alpha for alpha < lambda,
when lambda is a limit ordinal
L = union of all L_alpha

where Def(A) = the set of all sets X for which there exists a formula
P(y, x1, ..., xn) in the first-order language of set theory and a1,
...., an in A such that X = {a in A | <A, epsilon> |= P(a, a1, ...,
an)}.


--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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