Who shaves the barber?

Wheat Wizard

02:18

@Dennis Ok there is one thing that is still bothering me,
> There is no definition of set; we just know that all sets have to satisfy certain properties.
This seems to imply that the axioms are just a list of properties that things that have are sets. And most of them appear to be this typically starting with a forall. However when I started talking about throwing out regularity you also said:

> Well, even without Regularity, you cannot find an example. Otherwise, ZFC would be inconsistent.

It seems to me that without regularity our set a := {a} should fulfill all of the properties of a set and thus should be a set.

I don't mean to nitpick what you said but I am a bit confused about what the axioms of ZFC say

Dennis

a = {a} is not a definition; it's an equation. The only way of defining sets is specification schema, which doesn't allow you to use the term you're defining in the definition itself.

Whether there exists a set a such that a = {a} is an undecidable problem without Regularity, assuming ZFC is consistent.

And even if we assume its existence, there's still the problem of uniqueness. If there were several, which one are you referring to?

Finally, note that none of these axioms exist in a void. By combining them, there are several ramifications. Neither Pairing nor Regularity alone imply that no set can belong to itself, while both together prove that no such set can exist.

Wheat Wizard

To put it in terms of group-theory. Groups are defined by 4 properties. In plain english:

1) They are closed under a binary operation
2) They have an identity
3) Every element has an inverse
4) The binary operation must be associative

If we remove 2) there are groups that do not have an identity, for sure. By not including that requirement the number of things that are groups expand. How are these axioms different from the axioms of set theory?

Perhaps a bad example actually

because without 2) 3) becomes ill defined

perhaps I should have removed 3) or 4) instead

Dennis

The group "axioms" for a definition; (G,×) is a group if and only if it satisfies these four properties.

The set axioms are actual axioms; all sets satisfy these X properties.

Wheat Wizard

02:34

ok so even if it satisfies all the axioms it might not be a set?

Dennis

The latter isn't and if and only if. While all sets must obey the axioms, it's not enough to not contradict them to be a set.

Wheat Wizard

Is there anything we know that is a set?

Dennis

In part, this is because of Incompleteness: no matter how many axioms you add to ZFC, if you don't introduce inconsistencies or unprovable theorems, there will always be a proposition that is neither true nor false.

@WheatWizard Yes. We know the empty set is a set (that's an axiom). Because of pairing, we can form all "natural numbers" using their von Neumann representations. Because of the Axiom of Infinity, the collection of all natural numbers is also a set. From there, we can define integers, rationals, real number, complex numbers, vector spaces, and pretty much everything else you can find in the wild.

It takes a bit more work, but we can also define Ordinals and Cardinals (that's actually rather easy with Morse-Kelley).

Wheat Wizard

Ok now for the complete opposite question, Is there any "thing" that it is incomplete whether or not it is a set?

Dennis

Well, without Infinity, we would not know if the collection of all natural numbers is a set.

With ZF, the Vitali set would not be a set (requires the Axiom of Choice). Of course, once Choice is involved, you cannot really speak of the Vitali set, only one of them.

Wheat Wizard

02:42

To get my recursive sets I need to do more than just negate regularity. I need to modify the axiom schema of separation.

Dennis

Note that however you rearrange your axioms, a := {a} will never become a definition. The definition of a cannot involve a.

Definition in the a := b format, that is.

If I knew more about whatever you're trying to design, I might be able to help. Chances are, sets are not what you're after.

Wheat Wizard

I was really enthralled with the vonNeumann representation and I wanted to make a programming language that could not represent anything but sets. No boleans, functions, or integers.

At this point I'm less interested in the language and more interested in exploring set theory

Dennis

:)

Wheat Wizard

Unfortunately the only course at my uni that seems to deal with axioms is abstract algebra, so I don't get to do this stuff very often.

Thanks for all the help.

Dennis

Without knowing more about how you intend to represent things with sets, I suspect it's possible to get recursion without irregular sets. Note that 0 ∈ 1 ∈ 2 ∈ ... ∈ ω, and ∈ is transitive when it comes to ordinals.

So you can have an infinite chain of sets, all of which contain the same set and share a common structure.

These chains get weirder after ω. Much weirder.

For your language as well as your general interest, you might want to look into Ordinals.

Wheat Wizard

02:54

Ok will do

not having to rewrite all of set theory to implement my language is a plus

Dennis

Certainly, yes. Frege started in 1884 and we're still not done.

Pavel

03:15

@Dennis How is it infinite? Each link in the chain represents a natural number, and no natural number is infinitely large.

No matter where you start you eventually reach 1 and halt.

Wheat Wizard

Not if you start at ω

Pavel

Or if you start at 1 and count up... I'm stupid.

Wait, how does ω even work?

Wheat Wizard

Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order. An ordinal number is used to describe the order type of a well ordered set (though this does not work for a well ordered proper class). A well ordered set is a set with a relation...

and then

Transfinite number

Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use. == Definition == As with finite numbers, there are two ways of thinking o...

And just for fun:

How To Count Past Infinity

►

Its very hand wavy and has at least one error

but its pretty cool

Pavel

Is that the Vsauce one?

Wheat Wizard

Yeah

Pavel

03:30

I've seen that.

Wheat Wizard

Ok good then

Pavel

05:09

*Loads into main menu, sees 20 FPS and flickering*
Oh boy here we go
*Loads into game*
*150 FPS, no flickering*
*Max settings*
Wut.

Kritixi Lithos

07:56

@WheatWizard That's got to be my most favourite video of VSauce's channel

Transcript for

Who shaves the barber?