@Dennis I want to ask what that is but I also have like 2 minutes and it seems complicated ._.

Oh looks as if it has been frozen

@WheatWizard Classes replace sets as primitive terms. A class is a set if it can belong to another class; classes that aren't sets are proper classes. Essentially, a class is set-like, but it can contain more items. The universal class contains all sets as members.

> 102 days ago

@Pavel Come back when you have two weeks. :P

Will do.

I'll bring coffee

@Dennis How do we resolve Russell's Paradox? Can classes contain other classes?

By definition, a proper class cannot belong to another class (that would make it a set), so Russell's paradox is resolved: the universal class is not a set.

(And neither is the class of all sets that do not belong to themselves.)

Ah ok. i see why these are convenient

Some set theories employ the Axiom of Limitation of Size: a class is proper if and only if it is equipotent with the universal class. That makes a lot of things very easy.

How big is the universal class?

@WheatWizard Not sure how to answer that. It is literally the biggest thing there is.

Oh yeah I suppose it would be

That idea seems stange

that there can be a biggest cardinal

It works out nicely though. The ordinal of the universal class is simply the (proper) class of all ordinal sets.

Doesn't the replacement axiom preclude a largest cardinal?

I have no idea what we're talking about

@WheatWizard A largest cardinal set. The replacement schema does only apply to sets, even in theories that employ classes.

Ok. This still feels weird to me.

Limitation of Size actually replace the replacement schema.

Oh so are they incompatible?

← 26 messages moved from The Nineteenth Byte

@WheatWizard No, the replacement schema is a corollary. Since only proper classes can be equipotent with the universal class, the image of a set must be a set.

Limitation of Size also implies the Axiom of Choice. It can also be used to prove the Well-Ordering Theorem in a rather straightforward manner.

Woah, how

Once you've constructed the ordinal of the universal class, since ordinals are well-ordered, you got a well-ordering of the universal class and all of its subclasses.

Oh ok that makes sense

Oh replacement is a corollary of the limitation of size.

that makes sense

so the cardinality of the universal class must be an inaccessible?

As inaccessible as they come.

Or is it larger than that. I don't know much about the inacessibles, they have their own axioms and stuff.

Woah

Ok this is a little much for now. I think I'll have to take some time to soak it up now.

Different definitions of inaccessibility mean that a cardinal cannot be obtained by applying different set operations to a certain set of cardinals. Set operations on any set of sets will always result in a set, not a proper class.

Yeah that makes sense.

I'm just a little thrown off by the "there is a biggest cardinal" part

It's a bit shocking coming from ZFC, but it's really useful.

And it gives the best possible answer to the question Who shaves the barber?

The barber is a woman.

Wait, Can't we say that there is a thing, lets call it a class' that is a collection of classes? the same way we made classes from sets?

Then wouldn't we get larger cardinals?

Sure, those are called conglomerates in category theory.

Do we need to abandon the limitation of size to deal with conglomerates?

You can always define new things that didn't exist before. Some axioms are akin to religion. There are still people that reject proofs that rely on the Axiom of Choice.

@WheatWizard As stated, no. Conglomerates are "backwards-compatible" with all axioms concerning classes, as they do not mention conglomerates.

Ok so its not so much a limitation of size as it is a limitation of size of classes?

Well, yes. In any "class theory", classes are literally all there is.

ok. So it doesn't necessarily imply there is a largest cardinal.

The axioms of Morse-Kelley or whatever absolutely imply that there is a largest cardinal. Once you add more axioms and different undefined terms, you have a different theory.

Ok I think I see.

There's no absolute truth in math. Only what follows from the axioms.

Yeah, I guess I was just a bit confused about the largest cardinal thing. I thought it was meant in the way that limitation of size implied that no larger cardinal could be constructed without the introduction of contradictory axioms.

Whereas it seems the limitation of size just implies that more axioms are required for larger cardinals.

And more undefined terms.

Everything's relative. Are you familiar with the theorem that a countable union of countable sets is a countable set?

Yes, we did that in foundations for analysis

You can't prove that without the Axiom of Choice.

Actually I think we proved that the countable union of a closed sets is closed

That doesn't make it false, just not true. It's not an universal truth, but a consequence of the chosen axioms.

@WheatWizard That is false in most topologies. The union of the singleton sets {1/n} for all positive integers n is not closed.

Oh yeah

It must have been intersection or open sets.

Arbitrary intersections of closed sets are closed. No need for countability.

I think the union of open sets requires countability though.

yeah it most certainly does

No, arbitrary unions of open sets are open. That's part of the definition of topology.

Ok hm.

I do remember using countability, perhaps it is not necessary. This was a while ago and the fact that I can't really remember what I was proving doesn't show well for my memory.

Ok well thanks for the help. I think I'll go to sleep now

I should do the same. 2 AM again...

appears we are in the same time zone :)

And have similarly fubar'd sleep schedules. ;)