/sci/ - Science & Math

>>12208917
I think you make it more complicated than it has to be. Set theory gives a reification of higher order properties.

What you call
>the existence of nonexistence.
translates better into the "existence" of the proposition that is true for every thing [math]x[/math], and in an equalitarian theory, you can choose [math]\neg(x=x)[/math].

What you then need is either the axiom that explicitly says that this always-false predicate can be reifeid, i.e. talked about as a "thing". That's the axiom of the empty set and I agree that it is icky.

Alternatively, however, you only need the Axiom of Separation Schema for any predicate (and [math]\neg(x=x)[/math] is one such predicate).

I imagine you can cook up a theory without emptyset by disregarding either of those, but I've never seen such a theory. I've never seen a theory that doesn't validate at least Separation for predicative predicates.

>>12208980
>If I have set A which is a subset of B, then there is x in B such that x is in A
You're just defining the word "subset" in a way that validated your conclusion.
I have nothing against this notion, but it's not the common definition.

>>11624276
I'm home and looking at the chapter.

Lemma 13.10 (vii) says if [math] F [/math] is [math] \Sigma_n[/math] , then to prove [math] F[/math] is [math] \Delta_n [/math] , we only need to show it for the domain. Btw. this kind of thing was what I meant with using the graph-like aspect of the function set, I just looked out for a hint on that.
Although the proof of that statement itself is awfully short.

And I suppose the [math] \alpha [/math] in [math] \alpha\mapsto L_\alpha[/math] is always just the ordinal (although the book is not super clear on that either, but the +1 in Definition 13.1 gives not much other options). And the first line of proof 13.12 tells us that this domain, [math] {\mathrm Ord} [/math], is even [math] \Sigma_0 [/math] .