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

Aatu Koskensilta wrote:
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.


If ZF has a model, then, it has a supermodel that is not a model, for example modeling ZF + "there exists any irregular set", where any axiom restricting comprehension in ZF would lead to at most a submodel of ZF.

That is where, if ZF had no supermodels that were models, then ZF would be complete, in itself, because every statement is within the thus universal model.

Then, where ZF has a non-ZF-model supermodel, it witnesses irregular elements, then, of the regular elements there, which comprise ZF, their collection is irregular, by Russell.



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.


Defined by its elements, satisfying for each set that the element-of operator returns true, V is defined to be the collection of 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.


When you say "the axioms are provable", that's unclear.



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)}.



Empty!

Regards,

Ross F.
.