aboutlogic:premises #06 | What Is a Set? A Beginner’s Guide to Set Theory

00:00:03: Hello, everybody.

00:00:04: Welcome to a new episode of our premises series and today we wanted to talk little bit about set theory rather basic introduction so that We are all on the same level and can't have more advanced interviews.

00:00:18: And maybe the very first question and I will want to hear your take Torsten What is said?

00:00:25: what's the notion?

00:00:26: who introduced it?

00:00:31: Maybe I should first say that some people seem to think they don't like that theory.

00:00:38: Well, i'm actually a big fan in reality and maybe we can do these premises to clarify the situation.

00:00:53: so... I think sets or the idea of collection is quite old.

00:01:00: You talk about serial numbers, integers and groups... This was a meta-mathematical notion.

00:01:12: people talked about this but they didn't view sets as mathematical objects.

00:01:26: who changed this was Kantor, who started to consider sets as mathematical objects and like that you can talk about sets.

00:01:41: Like he talked about numbers yeah?

00:01:43: And I think that's his main contribution.

00:01:47: And here I mean Kantor had a particular idea of the set.

00:01:52: He said that a set is basically the same as property.

00:01:58: You can talk about green, if you have the property being green then it conforms to all green things.

00:02:05: so every property or every predicate leads to a set.

00:02:09: and what do you know about Kantor?

00:02:15: And we first want to remark on something you mentioned, that this is like an interesting quirk of mathematics and foundations in general.

00:02:22: That we take these meta notions and bring them into our very own theory right?

00:02:28: It's the same with proof theory.

00:02:30: In particular if we have limit results there couldn't be a proof or some thing not because it was wrong but independent.

00:02:40: There you need to make things precise, right?

00:02:43: You cannot wave around and say oh this is independent but you really need a notion of proof.

00:02:50: A notion of model or whatever.

00:02:52: And then we have an interesting quirk For the history of set theory.

00:02:56: I mean i'm no expert there But Kantor often discusses at the founder.

00:03:04: I mean historically they are like some discussions between priority, Kantor or Dedekind.

00:03:11: I mean there were a lot of logicians working on things.

00:03:13: but set theory proper is probably indeed started from Kantor and maybe not to make things already too complicated.

00:03:23: But the question what does it say?

00:03:25: It's philosophically very strongly debated.

00:03:29: one nowadays The iterative conception.

00:03:33: offset is their classic notion offsets.

00:03:36: you have The first level you have the empty set and then your iterates, this is containing the empty sets.

00:03:43: You construct more and more via power-set but it's not so clear that there are already Cantor's notions in there.

00:03:50: It seems to be just a collection of things.

00:03:56: I think Cantor identified sets with properties.

00:04:01: There was some argument whether he really did because Every property gives rise to a set, that's an important question.

00:04:09: But basically yeah... A property is raised to the set and I think in an important idea which you allude already we don't actually... I mean Kantos had all sorts of sets was very liberal but In the end as you said We only need the sets from the empty set.

00:04:34: And then we make new sets, so a set as property of sets.

00:04:42: in a way you can say and yeah an important question what Kantor I mean with sets?

00:04:52: You have...you talk about infinities.

00:04:56: obviously We all know finite sets and there are infinite sets and this simplest one is zero, one two and so on.

00:05:05: And there are a number of infinite sets.

00:05:09: I think the first question which Kantor wants to answer is whether all infinite sets are the same?

00:05:19: That's maybe naive if you could say, oh why does he have an infinite set?

00:05:23: As Kantor can show that it isn't the case!

00:05:27: He tried to compare the continuum of a set of real numbers with natural numbers.

00:05:35: And he had some argument why the continuum is larger than that of natural number, so there's no embedding from the natural numbers into the real numbers which are subjective.

00:05:55: and I think we talked about Hilbert's Hotel already.

00:05:59: That's not the case for lots of other sets.

00:06:01: For example, a set of even numbers is same as natural number.

00:06:09: So there are these infinities and they're all the same right?

00:06:13: Or I think we looked at this sequences of natural numbers.

00:06:18: it was the same as normal numbers so that has no increase.

00:06:24: but then real numbers increased in size And we already discussed another example, which I wanted to tell again.

00:06:33: But maybe people will remember this already?

00:06:36: This is the power set and yeah... So i had used this Hilbert hotel.

00:06:47: We looked at occupation lists for Hilberts Hotel Place where the rooms are number zero, one two and so on.

00:06:59: And an occupationalist says which room is occupied or in which room it's not?

00:07:05: Then we looked at the argument as a way to put all the occupation lists into the rooms.

00:07:19: We use this diagonal argument you can always find a list which is not in any room by like looking at Room Zero, entry zero and turn it around.

00:07:30: And Room One one if its occupied that's not occupied so on.

00:07:35: So we make new lists.

00:07:37: this list cannot be anywhere.

00:07:38: so thats basically intuitive version I would say of this diagonalization argument.

00:07:49: Yeah, I mean you said something and that is we construct sets.

00:07:53: And this was a point where the classical and the constructive mathematician need to be very precise what they mean.

00:08:00: because of course there are simpler constructions like uniting two sets since you get the members of both them.

00:08:08: but then power set it's big problem so-to speak.

00:08:13: It not so clear with all subsets or given sets And there is a lot of today.

00:08:19: It's pretty

00:08:20: easy for our final sets, so the set zero one two... So they are eight subsets right?

00:08:26: I mean you

00:08:27: can write them all down because for every element your answer to question if it's an element in this set or not its two times-two times two is eight.

00:08:36: but uh The set of all subsets on natural numbers Is very large and as we just have seen Its larger than the infinite set of natural number.

00:08:49: And indeed, the question is whether power sets actually... I mean when you learn this in class or school it seems to be a very clear and intuitive concept.

00:09:05: The set of all subsets of a set.

00:09:07: but if we look deeper then maybe developing some doubts that there's such straight forward idea right?

00:09:17: A tricky notion.

00:09:22: The step is made from the finite, which we understand and then you jump to infinite and say oh it's just the same but maybe its not You know?

00:09:31: And I mean there are many moments where you can see that one weird thing Is That the power set of natural numbers so much... ...that you cannot name everything anymore at least in a countable language We usually use.

00:09:44: So you cannot really point to every object living in these things and give it a

00:09:50: name.

00:09:51: And that's weird thing, right?

00:09:53: Because it feels concrete as you said but its not by the construction.

00:10:00: Maybe one step back to this constructive view on this iterate hierarchy or should we maybe introduce that idea now?

00:10:10: I mean there is standard viewing for today.

00:10:13: set theorist The classical one would be You can iterate this power set relation again and build up the universe of sets.

00:10:23: You already mentioned one step where this is very clear, like the finite set containing the empty set

00:10:29: etc.,

00:10:30: you often label these levels V zero for the empty ZV-one or the power zV two for that powerset etc.. And then...you can do it as often as you want!

00:10:47: those sets only contain finite sets.

00:10:51: And this is a valid model of set theory or large part-of-set theory, because it's not a model off there as an infinite set.

00:10:59: so we need to assume that right?

00:11:01: This isn't nothing only set.

00:11:03: theory needed to assume the classical type theory in the Russell sense also needed to assumed that for instance There are infinitely many types with infinite elements.

00:11:14: add that as a new axiom.

00:11:16: So you start with the empty set and then natural numbers if you want, so to speak And out of that You can construct everything by power sets more or less These.

00:11:28: iterative usage of the powerset operation will get tricky again because I mean if use though the infinite The infinites along all ordinals.

00:11:43: My plan was, I mean one idea would be to move two ordinates next right?

00:11:48: And this value ordering?

00:11:50: because okay so let's look a bit more at Kantor before we moved into axioms.

00:11:55: Because Kantor was before axiom were introduced actually.

00:12:02: Kantor once before formal logic was defined predicate logic wasn't there, so he was just waving his hands and explaining things clearly or in German obviously.

00:12:15: And that's it yeah... That is

00:12:18: interesting the shot to me.

00:12:19: you can either be clear our environment!

00:12:24: It was both clear

00:12:28: but another story.

00:12:32: I think he was particularly interested with.

00:12:35: you mentioned all the ordinals or well-orderings.

00:12:38: And so, well-ordering is a canonical example of less than relation on natural numbers because it's transitive.

00:12:51: but also when we start any number and always go down... You cannot go down forever!

00:13:02: zero, I mean you can't go down.

00:13:05: Any descending sequence is finite.

00:13:08: and that's also what was called an ordinary well ordering.

00:13:12: And there are much larger Well orderings.

00:13:17: so my favorite one it's called epsilon naught.

00:13:24: It can be illustrated by the Hydra game.

00:13:30: So there is this hydra, and it's like a horrible monster which has lots of heads.

00:13:41: It actually a tree finally branching trees.

00:13:44: in the end are all their hats.

00:13:46: And then Hackerus he can chop off head.

00:13:50: but this hyla is very terrible.

00:13:54: Each time when you chop off a hat, lots of hats can grow on lower levels.

00:13:58: Any number of heads can go.

00:14:00: so he's chopped off one head and one million heads grow but at the lowest level.

00:14:05: And now if we think about it You realize that eventually Hercules will win.

00:14:15: He would be able to defeat the Hydra.

00:14:23: This is also, I mean if you have all possible hydras.

00:14:28: So this a well ordering and it's very large order because that really much longer than the natural numbers And its related to power of arithmetic.

00:14:42: so... A

00:14:44: few with

00:14:44: audience records..

00:14:46: I think Von Plato mentioned ordinal of the Genson proof.

00:14:55: And now Kantor's question was whether even set could be well-ordered?

00:15:01: He had this principle, he said yeah I mean somehow from a set there should always be way to order elements with well ordering.

00:15:12: and what is the Well Ordering Principle?

00:15:15: and it's closely related.

00:15:17: And we had the axioms and it was shown that this is equivalent to the axio of choice.

00:15:24: But, but its very strange because if you think about how do well order the reals?

00:15:29: You know can define an order on a real.

00:15:33: so when you go down finally many steps If your have less than relation then doesn't work Because you could get zero point one zero point zero one can go down infinitely many, each number is smaller than the previous one.

00:15:49: I mean okay let's take us to positive areas.

00:15:52: so that doesn't work.

00:15:53: and the question is can we fix this?

00:15:55: Can you have an order on the reals which goes down always finite amount of time?

00:16:02: and Kantor just said yeah i think there has to be some such a thing when he had some sort of arguments why they don't need.

00:16:13: Yeah,

00:16:15: I mean that's another of these concrete non-concrete things.

00:16:20: right because we know there is one but We also know nowadays with our modern lens.

00:16:25: That these things are like highly undefinable But you cannot write any well ordering down and this Is i mean You might call it a buck?

00:16:34: I'm not sure about to do that.

00:16:37: for the classical mathematician There's an interesting feature right that we have yes undefined ability Proofs and we can go up a hierarchy showing how complex is an object.

00:16:49: And you can show that thing.

00:16:51: I find it quietly a bit strange if you say yeah, there's a well order on the real number in the same year and what is it?

00:16:59: Can't give you but It has to be one here

00:17:03: with everything i can construct.

00:17:05: just take your Choice function.

00:17:09: Take this one element out This Is The smallest then Take the next One Out This Is the second Smallest and do That for the whole length of the continuum that you're done.

00:17:19: It's very concrete, right?

00:17:20: I mean it's not a proof that the continuum is countable because i'm not done.

00:17:24: after omega-many steps there are still a lot there but after doing it for the hole length of The Continuum to the length of corresponding ordinal so just speak i am done.

00:17:36: yeah and Of course

00:17:37: its highly dependent on your choice function And I don't have one here Right now.

00:17:41: this That we can use To make it more Concrete.

00:17:45: Yeah, so that shows that using the axiom of choice you can construct a value order on the real set and do that.

00:17:56: But you cannot define it concretely.

00:17:58: but that's the problem.

00:18:00: is an abstract existence which cannot be satisfied

00:18:05: right?

00:18:05: I mean maybe too tiny remarks.

00:18:07: how messed up this whole situation is then?

00:18:11: we hardly know anything about of the continuum in classical set theory.

00:18:19: It can be nearly every cardinality, which is again these different sizes of infinity and I think it could even be class size principle but i'm not entirely sure how this models look like there.

00:18:36: you need some tiny tweaks... But we just know that they are a few things where theorem that says it couldn't be omega or something like that because of cofinality reasons, but we can outline just a handful of cases so to speak and then could everything from the second largest as smallest infinite cardinal too.

00:19:00: Something huge.

00:19:02: Okay yeah you should talk about the continuum hypothesis maybe later which is related.

00:19:08: We can't believe

00:19:12: it's the second card in

00:19:14: L.C.,

00:19:15: right?

00:19:15: But that's done for a little later, but yeah what was your plan and did you want to start with?

00:19:19: Yeah

00:19:20: my next view idea is talking about Frege.

00:19:23: So Frege... they're all German.

00:19:25: I mean these are all German mathematicians.

00:19:30: And so Frege he wanted He was actually on our line, he wanted to provide logical foundations for mathematics.

00:19:42: But... ...he wrote this Begriffschrift and I think he really invented predicate logic to do this right?

00:19:54: And he didn't exactly refer to Kantor but sort of had a kind-of set theory which is called extension.

00:20:04: And here it's this principle, which I mentioned before and I assigned to Kantor but actually it was Frege.

00:20:17: Namely that every property gives rise to an extension of a set.

00:20:21: This is full comprehension right?

00:20:23: If you have the property... You have a set!

00:20:28: I mean, in Frig's ontology there are these two levels.

00:20:31: Right?

00:20:31: There're this basic objects and then you have... We can think about properties-I think literally it were functions.

00:20:39: but functions and their characteristics function of a property that like there is duality there.

00:20:47: let's Think About Properties.

00:20:49: And Then This Principle You Mentioned The Principle Five Somehow Brought the Problem Because these properties corresponded to objects, in some sense breaking down this dichotomy of objects and properties.

00:21:07: And then give rise to a problem.

00:21:09: I think you will want it to mention now?

00:21:12: Yeah!

00:21:13: You know what comes next no?

00:21:16: Obviously... So Bert van Drasselow we leave Germany And he wrote a letter to Fregel.

00:21:28: I guess you know in which language the letter was written?

00:21:35: No, no!

00:21:36: The letter is also in German.

00:21:37: Oh okay...

00:21:40: It

00:21:43: wasn't aware of that.

00:21:46: He pointed out there's a flaw in Fregels system which is famously Rastlitz paradox.

00:21:57: And I always think it's a nice way to tell this, Which isn't my idea... Is with this barber here.

00:22:03: So there is a barber in the village and he puts his sign on the window and says I shave exactly all men or women.

00:22:13: All people should be careful!

00:22:14: All of them are in their villages.

00:22:16: you do not shave themselves.

00:22:17: I mean, okay.

00:22:19: We are living in old times.

00:22:20: we can talk about all the men only men shave and to Baba is also a man.

00:22:24: so that's maybe too fix to avoid any confusion And this cannot be right?

00:22:33: So he cannot shave All The People In The Village Who Do Not Shave Themselves Right?

00:22:39: I Mean.

00:22:40: we ask would he shave himself?

00:22:43: if the answer Is no He Apparently Would Because That On The Sign Yes, then he wouldn't because it's exactly those who don't shape themself.

00:22:54: So yeah... It was another of these self-references a bit like the Stagnall argument right?

00:23:01: And this actually you can relate to too and in case of the set theory what Drasso said or basically constructed as a set of all sets which do not contain themselves.

00:23:21: And that's the same argument.

00:23:23: if it doesn't, so its the same story as with the barber.

00:23:37: I mean there are many impossible sets right?

00:23:40: The sets for all ordinals cause similar problems because you unite over ordinals and still in ordinal and things like that.

00:23:47: So we needed to move on from this knife set theory at some point... In particular,

00:23:53: the audience will always not be well founded.

00:23:57: This is Burali Forty.

00:24:01: But so there was a crisis of set theory And it was depressing situation for Frege because he didn't really know.

00:24:15: There was no easy fix, basically.

00:24:20: Do you know what he did?

00:24:20: I mean... He put some note in his book or something like that...?

00:24:24: He had a very arrogant reply in some sense because he has the logical step of ''I cannot fix it, thus nobody can fix

00:24:32: it.''

00:24:33: Which is slightly optimistic about his faunal skills.

00:24:42: But yeah, the book was already in print when he received this letter from Russell and his reply went like oh thank you for destroying everything nobody will ever be able to repair that.

00:24:54: And then some sense.

00:24:55: I mean they are...I think we mentioned there were two diverging paths of repairing it.

00:25:02: one is the Russell type theory which wants to get rid off these two levels but have infinitely many Levels.

00:25:12: type of collection off types is second-order type collection.

00:25:15: Of those types the third order type and The other reply would be to collapse both levels and have oh everything as a set.

00:25:23: And then

00:25:25: okay, so I thought we answer the question How sets you can be fixed?

00:25:32: Yeah This is a bit of a dramatic in the next premises right

00:25:38: sounds great.

00:25:40: Even see all around in two weeks to finally learn whether set theory survived or not.

00:25:46: we will find out.

00:25:52: Maybe next week, We can already announce that a good friend of yours Will be our guest right?

00:25:58: Mike

00:26:00: Schulman

00:26:01: Yeah would you want some more pitch?

00:26:03: because your working together.

00:26:07: He is incredibly clever and interesting mathematician who started with sort of algebraic topology, related topics.

00:26:20: And then he became very well known for Homo Tropic Theory.

00:26:26: He's one of the driving forces behind this.

00:26:30: We are going to talk about a recent idea we have been working on together which is called higher observational type theory and that's a bit of a joke by Mike, because hot can mean two things.

00:26:47: It could be homotopy-type theory or higher observation-type theories to create confusion.

00:26:54: As always, careful!

00:26:56: Great then as always comment subscribe buy us some coffee become the channel member And thank you all for being here.

00:27:06: It's really a blast and a lot of fun.

00:27:08: then see you Thank

00:27:11: You.