Who shaves the barber?

Wheat Wizard

04:58

What really are classes? The only real explanation I got was from a professor that really doesn't like them.

Pavel

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

Wheat Wizard

Who shaves the barber?

Axiomatic set theory

Oh looks as if it has been frozen

Dennis

@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.

Wheat Wizard

> 102 days ago

Dennis

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

Pavel

05:00

Will do.

I'll bring coffee

Wheat Wizard

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

Dennis

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.)

Wheat Wizard

Ah ok. i see why these are convenient

Dennis

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.

Wheat Wizard

How big is the universal class?

Dennis

05:07

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

Wheat Wizard

Oh yeah I suppose it would be

That idea seems stange

that there can be a biggest cardinal

Dennis

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

Wheat Wizard

Doesn't the replacement axiom preclude a largest cardinal?

ATaco

I have no idea what we're talking about

Dennis

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

Wheat Wizard

05:13

Ok. This still feels weird to me.

Dennis

Limitation of Size actually replace the replacement schema.

Feeds

Dennis has unfrozen this room.

Wheat Wizard

Oh so are they incompatible?

Dennis

← 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.

Wheat Wizard

Woah, how

Dennis

05:20

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.

Wheat Wizard

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?

Dennis

As inaccessible as they come.

Wheat Wizard

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.

Dennis

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.

Wheat Wizard

Yeah that makes sense.

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

Dennis

05:31

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.

Wheat Wizard

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?

Dennis

Sure, those are called conglomerates in category theory.

Wheat Wizard

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

Dennis

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.

Wheat Wizard

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

Dennis

05:41

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

Wheat Wizard

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

Dennis

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.

Wheat Wizard

Ok I think I see.

Dennis

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

Wheat Wizard

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.

Dennis

05:49

And more undefined terms.

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

Wheat Wizard

Yes, we did that in foundations for analysis

Dennis

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

Wheat Wizard

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

Dennis

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.

Wheat Wizard

Oh yeah

It must have been intersection or open sets.

Dennis

05:56

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

Wheat Wizard

I think the union of open sets requires countability though.

yeah it most certainly does

Dennis

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

Wheat Wizard

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

Dennis

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

Wheat Wizard

appears we are in the same time zone :)

Dennis

06:08

And have similarly fubar'd sleep schedules. ;)

Who shaves the barber?

Transcript for

Who shaves the barber?