Re: Models and consistency

On Oct 31, 5:53 pm, "pers...@xxxxxxxxxxxxxx" <pers...@xxxxxxxxxxxxxx>
wrote:

I thought we clearly prove that L is a model of ZF.

Prove from WHAT axioms? Not the axioms of ZF alone.

And, to be definite, I guess you mean <L membership_on_L> is a model
of ZF. I.e., L is a universe for a model.

Two things: (1) L is a proper class, but a universe for a model is a
set, at least in ordinary model theory. (2) What we do hear about is
that the theory ZF+"exists inaccessible cardinal" proves that <L_k
membership_on_L_k> is a model of ZF where k is the least inaccessible
cardinal.

L satisfies all
axioms of ZF. This is often proved first before showing that continuum
hypothesis is true in L.

The continuum hypothesis is a theorem of ZF+"V=L".

But from what OTHER axioms do you prove that L satisfies the axioms of
ZF? And how do you define 'satisfies' where the universe is a proper
class and not a set? Are you using NBG for your model theory then? Is
it just as easy to prove "exists model (where the universe can be a
proper set) -> consistent" in NBG as to prove "exists model ->
consistent" in Z set theory? Have you checked it?

So, I thought there was no doubt that L is a model of ZF.

Doubt or lack thereof is a psychological or epistemological matter.
I'm addressing what we prove or don't prove.

I didn't
talk about seeing as in intution. I really thought L is a model of ZF
as L satisfies all ZF axioms.

But do you think that on the basis of some proof, and, if so, proof
from what axioms? Or do you think that on the basis of certain
INformal mathematical and/or philosophical considerations?

Well, guess I will get back my text books for a while.

What books and what sections in them are getting your notions from? I
too need to learn more about this subject. Perhaps if I see where
you're getting your notions I can better address them while I learn
something too.

MoeBlee
.

Relevant Pages

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