« first day (20 days earlier)      last day (20 days later) » 

spaceisdarkgreen
spaceisdarkgreen
04:10
@LinearChristmas I think the weak points is still in 4 (e.g. why does $P(\omega)^M$ mean the same thing in $M$ as is does in $V$)
Like you're saying "ZFC proves this countable transitive model of ZFC has a countable powerset and countable omega_1." Sure. Relativizing that to a countable transitive model M, M thinks "this countable transitive model of ZFC has a countable powerset and omega_1"... but why should "this countable transitive model" be M?
You're fooling yourself into thinking it is by saying that M is the model referred to by ZFC (which as we've mentioned, can probably work) and then saying that M somehow stays fixed when we relativize to M (this part won't).
(And as I think you mentioned before, it's absurd M would be the model M is referring to since M can't talk about itself.)
 
7 hours later…
Linear Christmas
Linear Christmas
11:49
I will try to evaluate point 4 in isolation; pretending not to know that contradictions eventually arise. It seems to me... that point 4 should hold in the stated form. This is due to the definition of relativisation, and due to the absoluteness of the concepts involved for (large enough) transitive $\in$-models. I will just say “absolute” from now on for brevity.

The formula $\varphi$ is of the form $\exists f\, \psi(f)$, so its $(M,\in)$-relativisation $\varphi^M$ is, by definition, $\exists f \in M\, \psi^M(f)$. Relativisation distributes nicely over conjunctions $\wedge$, so we must ge…
@spaceisdarkgreen
 
5 hours later…
spaceisdarkgreen
spaceisdarkgreen
16:30
@LinearChristmas No, same issue. P(omega)^M is not just some set... it's a definition in terms of the definition of M, whatever that is. dom(f)=P(omega)^M is a complicated abbreviation, not the statement that dom(f) = X where X is some set in M.
In other words, there's more relativization that needs to be done... It should be dom(f) = (P(omega)^M)^M...
(And now that I think about it, i'm more skeptical about the idea of defining M working in the first place, since I thought through a few simple examples and there always seems to be a snag)
Linear Christmas
Linear Christmas
17:34
@spaceisdarkgreen I agree with you. I do admit I am still confused about "objects" and formulae for which we can prove a unique object exists for. But if what you say is true, I am struggling to see the usefulness of absoluteness results as a whole...
Things seem a lot more rosy when I read textbooks on these things, I guess
@spaceisdarkgreen That being said, why would relativising something twice in a row would be any different to relativising just once. In the end, $(M, \epsilon)$ will get down to atomic subformulae, and along the way, the quantifiers get bounded to $M$. If I have bounded them once, bounding them again must surely be equivalent? Even predicate logic equivalent (if there is an $\in$, anways). After all, $\forall x \in M\, \Psi$ is amounts to saying the same thing as $\forall x\, (x \in M \Rightarrow \Psi)$
Doubling that, we simply get $\forall x\, (x \in M \Rightarrow (x \in M \Rightarrow \Psi))$ which is equivalent to $\forall x\, (x \in M \Rightarrow \Psi)$. Right?
And similarly for $\exists x$.
Of course, we would go deep inside $\Psi$ as well, but all we are really changing is the quantifier business when relativising to $(M, \in)$.
-----
I would understand, of course, from a meta-theoretic model theory perspective... Say we have a set object, let's say it's $S$. If we interpret it in a model, we would get $S^M$. Say that is again a set object (part of the original universe). So it too would have an interpretation in the model, $(S^M)^M$, which is not necessarily identical to $S^M$. But here we can say we are actually doing formal syntactical things...
@spaceisdarkgreen Could you provide one, simplest :), example of what you mean and the snags there?
(But I say again, if this is becoming tiresome, feel free to opt out. There will be no hard feelings, quite the opposite actually. It'd be best if you were my neighbour.)
spaceisdarkgreen
spaceisdarkgreen
18:19
@LinearChristmas Well, like for instance if you let M be the minimal transitive model of ZFC, you can't relativize that concept to M since there is no minimal transitive model of ZFC in M. Far as I can tell, same kind of thing happens when you try to use fragments and reflection (i.e. the minimal V_alpha satisfying some fragment), or formal introduction of a constant (as in ZFC+)

« first day (20 days earlier)      last day (20 days later) »