1. ## Re: All Sets

Originally Posted by Deveno
the axiom of regularity "stratifies" set theory
That is said more exactly, as I did in my previous post: The axiom of regularity provides that each set is in the cumulative hierarchy.

Originally Posted by Deveno
it does prevent us from the paradoxes we would obtain by "mixing levels"
Again, no, that's wrong. Axioms don't prevent paradoxes. It is by REFRAINING from certain axioms that we get by without contradiction.

Originally Posted by Deveno
What the original poster has shown, is that there does indeed exist a limit cardinal that is not finite
There is no finite limit ordinal anyway. And the existence of a limit ordinal falls right out of the axiom of infinity. And the poster didn't show anything. He's confused.: There DOES exist an enumeration of the sets he mentions.

Originally Posted by Deveno
the "limits of all the limits"
There's no such thing in ZFC.

Originally Posted by Deveno
Using some strange technique called "forcing" apparently one can conceive of cardinals which are "strongly inaccessable"
We don't need forcing merely to conceive the notion of strongly inaccessible cardinals. The notion is definable in rather simple set theoretical terms. As to existence: To prove that there exist inaccessible cardinals requires an axiom beyond ZFC. No forcing or anything else will provide the existence of inaccessible cardinals in ZFC; only by extending the ZFC axioms do we prove that there exist inaccessible cardinals.

2. ## Re: All Sets

I am sure you have thought about these things more deeply than me. Some of it, however is mere semantics: naive set theory (as far as I understand the history) led to some rather problemmatic paradoxes, which were remedied in stages by modifying the axiom system in use (proposed?).

As I see it, the axiom of infinity puts "limiting behavior" into set theory...the most striking, and perhaps important, example of this is "proof by induction" (which apparently has the same strength as the axiom of infinity). I agree that the original poster has some confusion about enumeration, which is quite common: people think infinite numbers or objects should behave like finite ones, only "bigger", but this is not so. The "acceptance" of "completed infinities" into mathematics was somewhat of a begrudging one...in fact, "proof by induction" is only valid because we deem it so, we cannot derive it from the other properties of ZFC, nor point to the natural world for empirical validity.

I certainly did not mean to imply ZFC asserts the existence of inacessable cardinals, just that the same "extension process" from which we pass from the finite to the infinite can be "turned again on itself" (repeatedly). That is, we can form a heirarchy of cumulative hierarchies (of which ZFC forms only the first), and then once again (and again....). I believe at some point we run out of names for these strange beasts.

As I understand it (poorly, at best) forcing is used to show "equi-consistency" of ZFC+other stuff with ZFC. Certainly you are correct that these methods are not needed to "conceive" of these notions...that was very poorly worded on my part...I just wanted to express the idea that lots of stuff lies beyond the purview of ZFC, even if we can express these notions in set-theoretic terms.

I recall reading in a category theory text once that there would be no need (in that text) for anything beyond "ensembles" (the ensemble of all (proper) classes being an example). It is heartening to see such unabashed optimism amid such uncertainty (I said ironically).

3. ## Re: All Sets

naive set theory (as far as I understand the history) led to some rather problemmatic paradoxes, which were remedied in stages by modifying the axiom system in use (proposed?)
Yes, but the axiom of regularity has nothing to do with it. Naive set theory is inconsistent because of the schema of unrestricted comprehension. The way the modifications avoided inconsistency is by not including the schema of unrestricted comprehension and instead using a weaker schema. The axiom of regularity plays no part in this.

"proof by induction" (which apparently has the same strength as the axiom of infinity)
I dont' know what result you're referring to. There are lots of forms of inductive proof.

"proof by induction" is only valid because we deem it so, we cannot derive it from the other properties of ZFC
I don't know what you have in mind here. ZFC does prove all kinds of inductive schemas. Even Z set theory, and even without the axiom of infinity, proves induction on natural numbers.

we can form a heirarchy of cumulative hierarchies (of which ZFC forms only the first), and then once again (and again....).
I would put it this way, we can axiomatically extend the cumulative hierarchy.

at some point we run out of names for these strange beasts.
We run out of names for things (in the language of set theory) as soon as we get merely to the power set of omega. That's quite soon in the hierarchy.

forcing is used to show "equi-consistency" of ZFC+other stuff with ZFC.
Forcing was first used to show that ZFC+"not-axiom of choice" is consistent and that ZFC+"not-continuum hyphothesis" is consistent. Forcing since then has been used for many other consistency results. However, by the second incompleteness theorem, it is not possible to use forcing within mere ZFC to show the consistency of ZFC let alone the consistency of ZFC with any added axioms. [Note: I am not an expert about forcing, so I can't go very far in a discussion about it.]

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