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Mathematicians use logic to access a platonic realm of objects, and Foundations and paradoxes are used to orientate towards these objects. Zorn's Lemma is used to find the maximal element of an ascending chain, but an ultra filter Axiom may be needed. Mathematical logicians take a different approach, studying logic and foundations in the spirit of a mathematician, not attempting to reach a higher truth. Explore the power of symbols to access the platonic realm, the consequences of a formal system, and the possibility of truth remaining even if the foundations shift.

Filters are a type of subset of a set X that satisfy certain axioms, such as X not being empty, supersets of a large set also being large, and the intersection of two large sets being large. On a finite set, there are four filters in total, two of which are principal filters. On an infinite set, there are an infinite number of filters, however none of them are principal filters. Filters are used to identify subsets of a set based on certain criteria, such as largeness or cardinality. They were introduced in the Rising Entropy article and are useful for next week's seminar.

A diagram is drawn to represent the natural numbers, with the empty set at the bottom and finite sets above that. Filters are used to classify sets as large or small, with the trivial filter considering only the whole set as large, and the co-finite filter considering all sets with a finite complement to be large. An ultra filter is one where every set is either large or collage, meaning that for any subset a, either a is in the filter or its complement is in the filter. This is demonstrated by taking any set of subsets with the finite intersection property and constructing a minimal filter containing it. In the case of a finite set, the minimal filter is the set of subsets containing the intersection of all the things inside it.

Ultra filters are sets with either large or small complements. For finite sets, the principal filter is an Ultra filter. Non-principal Ultra filters contain the co-finite filter, which only contains infinite sets, and cannot be on a finite set. This is because the intersection of all sets in the filter results in the complement of the whole finite set, which is not in the filter.

Zorn's Lemma states that any chain of filters on a set X containing a co-finite filter can be upper bounded by the union of the filters in the chain. This union filter is a maximal filter, and thus an ultra filter. To prove it is an ultra filter, the contrapositive is used: if the maximal filter is not an ultra filter, then it can be made bigger by adding a set that it hasn't decided on, making it an ultra filter and thus maximal. This allows for the proof of the existence of non-principal ultra filters on an infinite set.

Ultra filters are used to form equivalence relations on objects and to play a role in truth conditions for relations and term forming. Zorn's Lemma can be used to create an ultra filter from a given filter that is not an ultra filter. The finite intersection property states that any finite intersection of sets from a set of sets cannot yield the empty set. Ultra filters have applications in topology, such as proving Tychonoff's Theorem and creating non-standard models. It can also be used to generalize ideas like all but finitely many points, and is used to describe convergence of a sequence with the co-finite filter.

Mathematicians believe in the independent existence of mathematical objects, which can be accessed through logic. Foundations and paradoxes are used to orientate towards these objects, but the power of the symbolic structure used to reach out to this platonic realm is uncertain. The journey of Foundations starts with the platonic realm and attaching it to symbols, as the completeness theorem prevents us from starting with the symbol realm. We have an internal sense of truth, similar to our senses of the physical world, which allows us to understand the platonic realm. Zorn's Lemma is used to find the maximal element of an ascending chain, but an ultra filter Axiom may be needed, as not every ultra filter is maximal.

The speaker suggests focusing on the consequences of a formal system rather than the meta theory, and bringing a presumption of consistency to the table which can lead to interesting conclusions about models. Even if the foundations shift, some version of the truth may remain and be embedded into natural geometric or topological questions, suggesting that the truth of a proposition may not be tied to any particular foundations. Mathematical logicians take a different approach, studying logic and foundations in the spirit of a mathematician, not attempting to reach a higher truth.

A filter on a set X is a set of subsets which classifies largeness, rather than openness. It must contain the whole set, and not the empty set. It is a finer notion of cardinality than just talking about the size of sets. It was introduced in the Rising Entropy article, and is useful for next week's seminar.

Filters are a type of subset of a set X that satisfy certain axioms. These axioms include X not being empty, supersets of a large set also being large, and the intersection of two large sets being large. An example of filters on a finite set is shown with a Hasse diagram, where all subsets of X are drawn. All supersets of a large set are also large, and any finite intersection of large sets is also large.

Filters are used to identify subsets of a set based on certain criteria. The trivial filter is the entire set, while the principal filter is all subsets containing a particular element. On a finite set, there are four filters in total, two of which are principal filters. On an infinite set, there are an infinite number of filters, however none of them are principal filters.

A diagram is drawn to represent the natural numbers, where the empty set is at the bottom and the singletons are above that. Above the pairs and triples are all the finite sets, until the infinite set territory is reached. The co-finite filter is then discussed, which is the complement of the finite sets and has the same shape as the diagram. It has all of n, which is the top of the diagram.

Filters are used to classify sets as large or small. The trivial filter considers only the whole set as large, while the co-finite filter considers all sets with a finite complement to be large. The principal filter takes all sets containing a particular element to be large. Between the co-finite and principal filters lies an area of infinite, but not co-finite sets, where a filter can make decisions about which sets are considered large.

A filter f is an ultra filter if every set is either large or collage, meaning that for any subset a, either a is in the filter or its complement is in the filter. This is equivalent to saying the filter is maximal. This is demonstrated by taking any set of subsets that has the finite intersection property, which can then be used to construct a minimal filter containing it. In the case of a finite set, the minimal filter is the set of subsets containing the intersection of all the things inside it.

Ultra filters are opinionated sets that are either large or so small that their complement is large. For finite sets, the principal filter contains half of the sets and is an Ultra filter. For infinite sets, the principal filter is also an Ultra filter but not interesting for creating Ultra products. Non-principal Ultra filters contain the co-finite filter, meaning no finite sets are included. These filters pick between even and odd numbers and cannot be described by the large analogy.

Non-principle Ultra filters contain the co-finite filter, which only contains infinite sets. This means that a non-principle Ultra filter cannot be on a finite set. To prove this, consider a finite set of points, and take the complement of one of the points. This set must be in the filter, and when the intersection of all the sets is taken, it results in the complement of the whole finite set, meaning it is also in the filter. This contradicts the fact that the finite set is not in the filter, and proves that non-principle Ultra filters contain the co-finite filter.

Zorn's Lemma is used to prove the existence of non-principal Ultra filters on an infinite set. It states that any ascending chain of filters on the set X which contain a co-finite filter is upper bounded by some element of the Pro Set. This leads to a maximal filter, which is an Ultra filter.

A chain of filters, starting with F0 and ending with the power set of X, can be upper bounded by the union of all the filters in the chain. This union is a filter, as it satisfies axioms 1-3, and it is closed under intersection because of the subset relation up the chain. This union filter is then a maximal filter, and thus an ultra filter.

Zorn's Lemma states that any chain of filters on a set X containing a filter f has an upper bound, which is the union of the filters in the chain. This union is itself a filter and is the maximal filter. To prove it is an ultra filter, the contrapositive is used: if the maximal filter is not an ultra filter, then it can be made bigger by adding a set that it hasn't decided on. This new filter still has the finite intersection property, making it an ultra filter and thus maximal.

The finite intersection property states that any finite intersection of sets from a set of sets cannot yield the empty set. If a filter FBar contains some set a which is not in the filter and its complement is also not in the filter, then the intersection of the sets bi from 1 to n in the filter FBar cannot be empty. This shows that FBar union a generates a filter.

Zorn's Lemma can be used to create an ultra filter from a given filter that is not an ultra filter. This ultra filter is maximal and it is the only method for creating such a filter under the given conditions. This ultra filter can be used to construct a non-standard model, however, the filter cannot be concretely described as the only information present is the input given.

Ultra filter's role is to form an equivalence relation on objects and to play a role in truth conditions for relations and term forming. It can be used to generalize ideas like all but finitely many points, and is used to describe convergence of a sequence with the co-finite filter. It has applications in topology, such as proving Tychonoff's Theorem and creating non-standard models. It can be used to describe an alternating sequence of zero and one converging to one with respect to a filter that states all even numbers are large.

Zorn's Lemma was introduced to find the maximal element of an ascending chain. Applying Zone Slammer may not be enough to get the desired result, so an ultra filter Axiom may need to be added. Every maximal filter is an ultra filter, but not every ultra filter is maximal. It is not possible to use a magic wand to access the maximal filter, as it is simply a belief that it is there. The first four filter axioms can be satisfied, but adding the fifth axiom may create an internal inconsistency.

Mathematicians view mathematical objects as having a platonic existence, independent of any formalisation or foundations. Logic is used to reason correctly about these objects, and while different foundations exist, they do not affect the true nature of a mathematical object. Foundations and paradoxes are used to orientate towards these objects.

There is a belief in the independent existence of mathematical objects, but the power of the symbolic structure used to reach out to this platonic realm is uncertain. To think of the symbolic stuff as being equal to the platonic Transcendence stuff is to miss the point. The journey of Foundations starts with the platonic realm and attaching it to symbols, as the completeness theorem prevents us from starting with the symbol realm. We have an internal sense of truth, similar to our senses of the physical world, which allows us to understand the platonic realm.

Foundations of mathematics can be studied in a mathematical way, rather than trying to construct a logical system from symbols on a table. This project is often viewed as overly simplistic and not a reliable guide to the Platonic realm. Mathematical logicians take a different approach, studying logic and foundations in the spirit of a mathematician, not attempting to reach a higher truth. Discussions of foundations can become vague and lead to less understanding, so it is important to acknowledge that the questions asked are still motivated by the same quest.

The speaker suggests zooming in on consequences of a formal system rather than zooming out to the meta theory. They agree that bringing a presumption of consistency to the table is fair, and that this can lead to interesting conclusions about models. They suggest that even if the foundations shift, some version of the truth will remain, and feel invested in its reality beyond the foundations.

Model theory and algebraic geometry can be discussed in the same way, but there is a lingering sense that set theory and model theory may not be "real" in some sense. However, this intuition is often refuted when set theoretic issues are embedded into natural geometric or topological questions. This suggests that the truth of a proposition may not be tied to any particular foundations, and that it is possible to shift from one set of foundations to another.

the goal of these two seminars so today and uh next week is I want to uh introduce to the foundation seminar a method of actually getting a uh somewhat more concrete uh example of a non-standard model I guess there's been several results uh already which tell you okay like there's this model that you intended but we can also do some like trick with the language to like force there to be more objects than you wanted um in the model of your theory or less objects like with the downward enough with lower low and homescholum theorems or also the incompleteness theorem sort of uh saying that uh like it's a that's another way of pointing to the existence of like these weird no what are called non-standard models um but there's no it doesn't actually sort of give you any sort of grasp as to what an example of a model could be and like sort of what do the objects look like in it and sort of why does this non-standardness occur um and so that's what I want to get to um and uh in order to sort of achieve more in these two lectures today is basically going to be uh doing your like doing the grunt work um and so it might feel a bit unmotivated I'm going to be talking about filters and Ultra filters and so for now I'm I'm kind of just hoping you'll accept uh just accept filters as a interesting Gadget that is not necessarily like completely motivated thus far but they'll be useful for for next week uh one sec let me grab my pen okay sorry first definition uh what is a filter so a filter um muchla is sort of in structure it's much like a uh a topology on a set but it has different axioms and the idea is to instead of classifying openness it classifies largeness um and and it's sort of a finer notion of like what which sets are considered large uh than like just talking about cardinalities of sets um so and we'll return to that idea in a second um so yeah filter is on a set X is a subset of uh is a set of of subsets of X which decides which subjects should be considered large so let me write that do you know who francophone filters uh no idea oh I just came across them in the the rising entropy article which oh there wasn't some sort of Vlog series that uh is linked in the notes besides which subsets and here's what here's sort of the the rules that it needs to satisfy so first um the whole set is large so you have to have the the whole set itself in the filter secondly the empty set is not large so I guess these these are the two like least controversial statements you could make

about largeness um maybe the first one is a little controversial if like I don't know if X's are just a very small finite set um does this implicitly require X to not be empty um yeah because then the empty set can't both be in it and not in it yeah so a filter is another episode yeah well I don't have to there's just uh yeah or you've left it implicitly there that's fine got it yeah well as long as you satisfy these the first two axioms like you're not gonna X can't be an empty set anyway [Music] um Okay the third one is about supersets so if you start with a large set and then you find some superset of that set B uh then B is also large whoops not subset uh B is in F so therefore like also uh unions of of large sets are also lodge because you're just creating a superset um so it's a yep um and then finally if you have two large sets the intersection is also large um and so therefore any like finite intersection of large sets is also large um I'm going to give an example now which will probably we can play around with to sort of classify the filters on it um just to get sort of more familiar with filters um but this is going to be the filters on a finite set which for the purposes of next lecture kind of completely uninteresting um the more interesting case is when you have X being an infinite set um but uh let's just understand this first so I'm going to draw a hassa diagram which is just a neat way to visualize what the pot the the pro set of subsets of X looks like so at the bottom we've got the empty set then just above that there's three Singletons above that there's the pairs sorry Valley and you've switched to zero one two anyway X is one two three so yeah before you write all of them yeah and you actually use zero and two thanks now one two um okay and I'm gonna just fill in the sort of uh edges that describe subset relationships when you read them upward now drone is a bit compact but I hope I mean if you've seen this before that you already know that there's a there's a badly drawn Cube here when you look when you look at the edges um but that that can be useful for intuition later um so so what are some example filters I can pull out of here um so I to be clear I've drawn the I've drawn all of the subsets of X and we're going to pull it like figure out uh which of these subject subsets can we collect into a filter that satisfies all the axioms so um a way to proceed in light of the third Axiom is okay uh whenever you have a large set all the supersets are also

large so you need to like you might as well just start from the top and sort of go down slowly and add sets as needed so you can be very lazy and just have the the whole set as large and everything else is small and you can easily verify that satisfies all the axioms there's no like other supersets and there's no interesting intersections going on so that's a filter that's called the trivial filter um then uh you if we travel down let's just add one more a lot one more uh set to the filter and see if it's still a filter um and in fact if we just add 0 1 and there's there's still no more interesting intersections um you still just get one of these two sets and there's no more supersets um and so clearly so that's also a filter and there's two more that are like that when you if you pick the other two pairs uh now uh if we try and introduce an interesting intersection say if we pick uh if we try and put uh let's put we've got zero one two in that let's put zero one and zero two uh what happens by Axiom three you have to also throw in uh the set with just oh whoops uh sorry not that's it the set was just zero in it because that's where that's the uh that's the intersection of those two sets and you'll find that this actually gives you the set of all subsets of x such that zero is in the set and we call these filters principal filters because that that characterized by like you just say all this all the subsets which contain this particular element in mind um and again there's two more like that there's two more principal principal filters I.E um all the you have all the super sets of the set containing one and all the supersets or the second containing two those are also two other filters um and is there any more um no there's not because uh if you uh I mean this isn't a complete proof but just for example if you start with one of the principal filters and you're interested in adding um I guess anything any other sets in the diagram so for example if you have the the one point set containing zero uh you can't add the one point set containing one uh because the intersection is empty um so that's that's actually all of the uh there's a there's all the filters on this set um and so now let's talk about the filters on an infinite set so I'll go over to the other board here I've been surprised by the four facts in can you say something more about it I just don't see why you would expect the intersection of two being said to be big yeah so um I I'm pretty sure so when we do the uh

I guess I'm trying I'm sort of in absence of anything else I'm kind of just motivating it by this idea of largeness and so I guess that's exactly what you're asking how do you sort of extract this condition out of the wine to classify logginess um I don't know there might be a way to do that uh like I think there's you can sort of uh there's probably some way to motivate the reason why you can't have uh infinite intersections and like you and so this is kind of just a choice to only allow finite intersections I think is probably the more uh probably gets closer to answering that question because you can have infinite unions because you can just have uh any supersets as far as I remember the original example of a filter is just the set of all neighborhoods of a point in a topology so then it's closed down behind the sections so I think um I think a better answer would be something more like uh I haven't fully worked out the details of the proof from next week but um when we like uh I'm hoping to just find the the need for this Axiom amongst like uh the inductive proof on the complexity of a formula that like involves conjunction or maybe it's disjunction or something will require the intersection to also be large yeah yeah I can yeah sure okay thanks Okay so let's talk about uh the case where X X is the natural numbers because that's the example we'll be interested in um so let me draw like try and draw the analogous diagram for natural numbers um so we've got n at the top empty set at the bottom and just like before above the empty set we have all the Singletons uh zero one two and so on there's infinite in many of those um above that you also have the pairs and then the triples uh like as as we go up where we're getting just all of the uh finite sets are being arranged here so you can imagine them sitting in little levels of your pro set um but they stop at some point and you sort of enter the infinite Set uh territory um and so I'm going to tell you about the co-finite filter here and so the interesting the interesting observation here is that uh is that the finite sets are not half of this diagram that kind of uh they're obviously a very small part of the diagram because of I don't know cardinality reasons but um what we're actually going to do is reflect this part of the diagram um to the co-finite sets up here which kind of has the same shape it's kind of the same data but it's it's a co-finite set is just a complement of uh of a finite set so it has all of n

except for finitely many points so it so it looks roughly the same let me just label some things here um and so here you have things like like just like you had the the single tins you also have the co Singletons up here like an example would be say n without zero um so there's all the co-finite sets and between them here is this like large scary territory of uh finite but I'm sorry infinite but not co-finite sets I guess you could also call them infinite and co-infinite sets um so both the set and its complement are infinite sets um so some examples are a set of even numbers uh the odd numbers um I don't know prime numbers um and this is the sort of part where the filter is free to sort of make decisions uh about like a filter could say that the even um the set of or even numbers is large but the set of all odd numbers is uh is not large [Music] um I'll come back to this when I talk to talk about Ultra filters but um I guess the example here is that uh the set the the the following stat is a filter um f equals the set of all uh subsets of n or it could be subsets of any set X really but I'll just talk about n such that X well not X n uh minus that set is finite so if you if you just decide to say only the co-finite sets are large like you're being very strict about like you're only like a large set if you have all like if you contain all but finally many points of the of the entire space uh then that's what the current finite filter says um okay I'll move on but we'll come back to this slide again in the future um oh I guess I didn't sort of talk about just verify that the axioms are true of it so um clearly X is co-finite the empty set is not carefinite uh supersets of curve finite sets are still co-finite and intersections are as well um I guess I'm leaving you to verify that mentally on your own um but the curve finite filter is kind of is is pretty like prototypical of the like kind of starting point of making an ultra filter um it turns out that all the interesting Ultra filters contain the co-finite filter and making an ultra filter comes down to making choices about uh which sets in this sort of Uncharted Territory uh also large um okay let's move on to the next section um okay so so there's three examples I gave across those two boards where trivial filter where you just take only the whole set is fine is large um then then you have the co-finite filter and all the principal filters um again this is the next limit I want to prove and this is just saying that if you

start with any uh set of subsets that has the finite intersection property then you can then there's a minimal filter containing it so the finite intersection property is just saying uh any finite intersection of the sets in your given set uh is not empty um so it's essentially just saying that you're not going to violate the third Axiom just by intersecting um finally many sets that you started with um and we'll use this Lemma at some point later so any subset s with the f i p finite intersections are non-empty has a minimal filter containing it and so I'll just give a sort of uh we can just start with we can just give an example which is um I guess in the example from earlier we started with the set 0 1 inside that three-point set and we also had the Set uh zero two and so say that this is our starting set s is is the set containing these two sets um if we want to make a filter out of this well we need to satisfy the Third uh was it the third accident the fourth one the the intersection one and so therefore we need to add in this set uh this one here and then we also need to satisfy the fourth one which is uh we need all the supersets which just is a matter of adding uh the whole set in this case but that's the the sort of two-step process to to constructing the filter really is uh first you close downward under intersections so take the closure of this set under intersections and then close upwards under supersets I'm not going to super detailed uh give a super div verification of the axons to this but that's the idea um so in a case when X is finite this minimal filter has an easy description right it's the set of subsets containing the intersection of all the things inside s yeah and in the yeah that's in a case like you're just describing it as the set of all supersets of finite intersections of yeah exactly yeah okay makes sense all right any other questions for now okay well now I'll Define Ultra filters um and I'll come back to some of my examples from before and talk about Ultra filters uh on those sets so a filter f is an ultra filter foreign if every set is either large or collage so concretely what I mean by that is the fifth Axiom that you can add to the collection which is for any for any subset a foreign you have a is is in the filter so it's large or x minus a is in the filter so another like definition of alt Ultra filter is that it's like it's it's maximal um and I'll actually prove that equivalence or One Direction of that equivalence later but this uh version is

just saying take any set you like um the ultra filter is like maximally opinionated on it in that it either says it's large or it's so small that the the complement of it is large um and so what you'll find is okay let's let's go back to some of the examples from before and talk about which uh Ultra filters um so the ultra filter condition in a way like especially for finite sets means you have like half of the sets um so if you can kind of see the cube on here um maybe I'll I'll sort of draw it again uh next to it the cube here looks something like this um and here actually let me redraw that like that uh here the prince of the principal filter containing the that sort of starts with this set here you can see it's it's sort of one face of the cube and it's exactly half of the sets so principal filters are Ultra filters but they're kind of just the boring ones um so in the case of the infinite uh starting with an infinite Set uh the principal filters are also Ultra filters here um but we'll find that like they're not super interesting for when you when you go to create Ultra products um you kind of just end up with what you started with and so the the real interesting uh Ultra filters are the ones that are on an infinite set like a set of natural numbers um but which aren't uh which on principle and what you'll find which is what I'll prove now is that uh not being principal actually forces you to remove all of the um all of if you're an ultra filter you can't have any finite sets and so what that means is you're actually you contain the co-finite filter um so so in this picture it means if you want an interesting uh principle sorry interesting sorry non-principle Ultra filter you can't just stop here um sorry uh let me restate that I guess let's look at what like a if you think about what a principal filter looks like on this sort of diagram here it kind of just contains the like half of the whole thing um like it's sort of dense in this diagram but you have like exactly half of everything at every level um of this diagram um but when when you have uh when you have a non-principal one uh it ends up looking a little more like this where you sort of include all the co-finite sets so that means you can't have any finite sets and then you have to kind of just get half of this stuff here and so that means picking between the even numbers and the odd numbers um and I think that's kind of where like the the sort of large analogy kind of breaks down is like who's to say that

the even numbers are large in the set of natural numbers and the odd numbers are small it's it's kind of a I know the the analogy doesn't really track there but um but let's go ahead and prove what I was just saying which is that uh if you want to be non-principle so you actually got an interesting uh Ultra filter then you at least have the co-finite filter and that's the next level sorry for going back and forth so much I hope you can keep up thanks so every non-principle Ultra filter contains the co-finite filter foreign okay let's prove that oh and you'll observe that uh this actually means that uh you can't have uh non-principle Ultra vilters on a finite set because the curve finite filter only has infinite sets in it um okay so consider uh so equivalently we this amounts to proving that uh F the the non-principle ultra filter in question has no finite sets so consider some some finite set a subset of the whole Space and let me draw a picture here so this is all of x this is your finite subset a and so there's finally many points inside it um and the observation is that if I take the complement of say one of the points uh what I mean is exactly this set here where a is one of these points in the finite set a and I so what that looks like is this so it's all the it's the whole Space except for exactly that point um this exactly because the the filter is non-principle and it's an ultra filter this set has to be in the filter because this this Singleton set is not in the filter um because if they were uh yeah like we need at least one of these in there uh either this set or its complement and this set isn't in there so its complement has to be in there but then note that if we take uh the intersection over all the elements in a of these large sets we just end up with um the complement of that whole finite set a so that's that's a way of just taking any finite set and showing that uh it copy in the filter because then we're contradicting that uh this is in the filter foreign any questions about that does that make sense wait sorry so why can't you have a and X plus a in the filter uh because then the intersection would also have to be in the filter and the intersection of course yeah okay yeah so it makes sense yeah that I was going to say before that the condition five is written as a sort of inclusive or but by the the intersection Axiom uh you can just read it as an exclusive or oh yeah that makes sense okay got it okay so um if we want so the the lesson of this is saying that

if we well I I've told you that the only interesting uh Ultra filters are the non-princivable ones and so this is telling you that to make a interesting Ultra filter you're going to want to start with the co-finite filter and then fill in the the the gaps of all the uh decide which infinite but not uh co-finite sets count as large and so uh now I'm going to prove that there actually exists um such non-princible Ultra filters on an infinite set and we'll do that over here foreign so this is called the ultra filter limo which says that every filter is contained in an ultra filter and so yeah as I was saying like say we're starting with the co-finite filter and we want to find some Ultra filter that it's contained in uh I was talking about that Gap that sort of infinite gap between the infinite sets sorry between the co-finite sets and the infinite sets and there's uncountably many sets there so I don't know what's what's everyone's favorite way of making uncountably many decisions um any any suggestions well we're gonna use we're gonna use zon's limo yeah so um the proof is play Zorn's lemon um I guess I find it funny the way like the sort of the way this is often communicated is like um I don't know that like Zorn's glamor is this gadget that's going to give me like this set in some way but I think it's really just sort of saying like we might as well just have a the ultra Phil dilemma be like an axiom and like on its own Merit we just sort of say like oh of course you can just sort of make the decisions um sensibly like I don't know just like just just make infinitely many decisions it's sort of the same uh I I guess we're kind of reducing this case of wondering whether Ultra filters non-principal Ultra filters exist too uh we've already decided in like anything that fits Zorn's Lemma we were like asserted it it to exist um sorry I don't know I'll we should leave that commentary to the side for the moment actually um okay so what do we apply zones limit to uh the pro set yeah exactly um so the pro set of filters on uh the whole set X which contain f foreign and so the the two steps of this argument are going to be um first uh justifying why we satisfy the premises of Zorn's Lemma which is that if you have a chain of uh if you have an as an ascending chain in this Pro Set any ascending chain uh that such a chain is upper bounded by some uh some such element of the pro set and then from there like out of Zorn's Lemma pops uh this uh maximal filter and then I have to

argue that a maximal filter is an ultra filter um so uh I could just like you can make the observation that all right given a chain of filters maybe I'll just write down what that looks like just so you have a better picture in your head chain of filters starting with f which I'm also going to call F0 which is a subset of F1 which is a subset of F2 and so on so the the chain might be countably infinite or um okay and so you can see that this the real the partial order on this Pro Set is the subset relation so it's a matter of every uh filter in this chain might be adding more sets that it considers large um okay and so of course all of these are upper bounded by uh the power set of X but um this is not an element of the preset the whole power set is not a filter itself so that's just making that observation isn't enough for zon's Lemma you need to when you when you say every uh ascending chain is upper bounded it needs to be upper bounded by something in the Pro Set so we need to identify some filter which is actually in the Pro Set that upper bounds this this chain um and in this case we can just take the the Union so their Union is a filter and so let me justify why um well firstly axioms one to three uh hold for any Union of filters so I hope you see that this is also not controversial so the first two axioms or that the whole set is in the filter and the empty set is not in the filter if you just take a union of a bunch of filters on X you're not ever you're not not ever adding the empty set nor removing uh the whole set so trivially trivially that's fine and also in the case of uh needing every superset of a set in the filter to be large you're also not removing uh any any of the supersets of the sets that you start with from any of the uh filters that you're unioning um so the all there is to check is really the the intersection part uh so now I'll argue that the Union uh over again this is the filter that I'm this is the filter which is the which is the upper bound filter for this this chain so this thing is closed under intersection because well if you take any pair of subsets in this in this big Union filter well they come from two of these um two of these filters or like they might be from the same filter but say for example a comes from F0 and B comes from F1 well the only observation you have to make is that because there's a subset relation going on here up the chain um a in this case because a was in a filter that comes before B we know that a is

also going to be in the same filter that b is in because of that subset relation between the filters and then by because that particular filter is a filter the intersection is also in that filter so therefore it's in the Union so I'll write that argument now because for any pair a B in the Union uh where let's say a is in FJ B is in f k and without loss of generality uh let's just say that J is less than or equal to K foreign because if if it's not then just switch I and J um we have a is in f j which is a subset of f k because FK appears appears later or at the same position as f j so it's a subset um so that means a is in FK so therefore the intersection um what am I saying here so a intersection B is in is nfk um which is contained in the Union so the intersection is also in the Union okay so that shows that take any training filters uh take their unit the union of those filters that that whole set of subsets is itself a filter and clearly that's an upper Bound for all of the filters in the chain so uh we've done half of our work now so by Zorn's Lemma let me spell correctly there is a maximal foreign filter by inclusion so when I say maximal such filter I'm talking about amongst the post set of filters on X containing f okay so now I just have to argue that what I've got is this maximal filter is actually an ultra filter foreign proof so we'll do this by sort of proving the contrapositive so if I have something which is not an ultra filter then I'm going to show that it can't therefore be maximal so you've got some filter and I'm going to call it F Bar because it's it's going to be uh I don't know F Bar is just the name I'm going to use for the maximal filter but this just holds for any filter uh so if a filter F Bar is not an ultra filter um then by not satisfying Axiom 5 you have some there's some set which uh the filter F hasn't decided on so you have a is not in the filter and X without a is also not in the filter and so the claim is obviously that what we're going to do is just make the this filter bigger and then claim that uh or we claim that we can make the filter bigger first by adding uh first by adding well we could I we could add either of these sets really um so let's add in uh the set a so we're going to call a large now and so the observation is that it has the finite intersection property and so how do you see that well uh firstly when we're talking about finite intersections of sets in this set of subsets we don't really care about the ones which

uh which don't involve a because we already know they're not going to give us the empty set um maybe I should repeat the finite intersection property states that any finite intersection of sets from your set of sets doesn't yield the empty set and so because F Bar itself is a filter um if you've got some final intersection that doesn't involve a then we already know that it's not empty so we just need to check that if you've got a finite intersection that does involve a then it's still can't be empty um and so uh let's consider any finite intersection involving a can be written like this it's like a intersection and then the intersection of uh a whole bunch of sets bi from one to to some number n uh with all of the bi in the filter F Bar um and so if I just draw a picture we're showing that uh we can't have this intersection be empty so uh so here's the whole set X is a if the intersection is empty then they're sort of mutually exclusive there's the intersection of the bis um but then if it's empty we'll then just observe that uh that the intersection of the bis then belong to the complement of a oops let me draw this part first so this is the complement of a all of that spaces is x minus a um so then we have the intersection of the bis is a subset of X without a but then observe by I think it was the is the fourth Axiom um by number four the final intersection Axiom we have bi the intersection of the bis is in the filter F Bar um and so by the superset Axiom we should have uh x minus a should be in uh should be in in the filter F Bar make sure you write this and by three uh that should also be an F Bar because it's a superset of this large set um but that can't be the case because we we just posited that um that X without a was in in the filter uh this should have said F Bar not f um there's no F involved here um and so that that produces a contradiction so that you could you cannot have uh if you sorry that we're not finished here what I've done is I've started with a filter which is not an ultra filter then I've shown that um I've shown that we if you have some some set a which is not in the filter and its complement is also not in the filter then what did I prove here I proved that this set here has the finite intersection property foreign it means that hence FBA Union a generates a filter that was what I proved earlier that as long as you've got a set of subsets that of of which you cannot create the empty set by taking finite intersections then you can

uh close it on the intersections and then close it under supersets and then you've got a filter um so F Bar Union the set containing a generates a filter properly containing uh F Bar so that means F Bar is not uh maximal so what I approved was that if you've got a filter which is not an ultra filter then it can't be maximal and so therefore if you have a maximal filter it has to be an ultra filter um and so therefore the the filter the maximal filter that Zorn's Lemma gave us um is therefore an ultra filter um so that completes the sort of technical part well the the work we did today I guess I want to just say finally that uh I guess you might be wondering okay uh under no conditions uh I've shown that you can use someone's Lemma but I don't know you could use zon's limit to I don't know prove that every Vector space has a basis but we do know how to uh we can just start with the basis in mind and we don't have to like uh use the axiomy of choice or anything but uh to my knowledge in this case like this is really the only method for like to create an ultra filter under these conditions um like I can't sort of I can't actually pick out just an example even if I know that the set X is uh the set of natural numbers or something um and so this kind of this should feel a bit disappointed disappointing if if like given my slogan at the start that I'm going to produce a concrete uh non-standard model using Ultra products which depend on Ultra filters um because I like there's this sort of Gap in creating an ultra filter well at least a non-principle one uh which kind of critically just involves like knowing of the existence of such things but like not actually being able to concretely describe an example so what is the input term that you'll put into this Lemma when you're constructing these non-standard models which is sort of the only information present then if the rest is just a wrapper that says do Zorn's Lemma or lead zones that might give you something sorry could you explain that again what do you mean yeah I can't ask a precise question because I don't know how you're going to use this I'm just curious in the cases where you're going to apply this to get an ultra filter in there then use that to construct a non-standard model of something or other is it going to be the case that the filter we start with is somehow concretely described and then we say take an ultra filter which contains it or is it even less direct than that where the filter the because we start with at least n we

we kind of know all right let me let me just say the ultra filter its role is in forming the uh it informs the equivalence relation on the objects that's the role in sort of like uh what the underlying set of the model is um it also then plays a role in in the truth condition for the relations and also the the term forming [Music] um but yeah that's where it sneaks in but I mean to my mind if you're happy to use Zone's Lima everywhere else in mathematics you should be happy to use it here um any other questions I think that's good um cool I also wanted to say this so in the uh the the sort of first reference in the node so distributed um there's some there's a fair few interesting applications of filters um so for example I don't know if you've ever heard of like some condition anywhere in mathematics that says something like uh like all but finitely many points blah blah blah then you can use filters to kind of generalize that idea uh so you can describe uh convergence of a sequence with the co-finite filter um I think it's something like uh the like a sequence converges to a given limit point if all the open sets the pre-images of the open sets um containing that limit point are all co-finite as in there's only finally many points that don't live uh in within that set is the idea of convergence but when you uh then sort of trying to when if you then redefine convergence but with respect to a filter you can use other filters and then say things like you can have like a sequence of like alt an alternating sequence like zero one zero one zero one and with respect to a filter that says that all the even numbers are large uh you can say that that sequence converges to to one so and there are other ways you can also uh there are other things you can do with it it sort of has applications in topology I think you can also prove tick enough's theorem um which I always found mind-blowing from metric and Hilbert's faces uh yeah and and of course you can use it to create non-standard models which is what we'll do next week um but that's it for me thanks again yeah yeah now now everybody can find it there okay you didn't like the the secrecy but I guess now that you know where it is that's right like now I know I like it yeah I was thinking of making some kind of like teleport system where like you walk into one tree and like pop out of another one um because basically as a solution to like at the moment I've kind of just been Shifting the spawn location um so that you like

so that you end up where we're going to be doing the seminar but I don't think that's uh uh I'm not going to be able to do that when we have multiple pockets and like different things are happening in different Pockets I don't know oh you can't get me uh not very easily yeah so the point I was making before is that whenever I like applied Zone Slammer or something I don't feel like I've uh like I could kind of be equally satisfied if I just sort of like added like an ultra filter Axiom sure like there's not sort of like much going on I'm just sort of classifying it as this tucks in the same set of examples that uh like Zorn's Lima was introduced because of like specific instances like this right where we wanted to like uh get some maximal uh element of some extent ascending chain um and so it's just sort of a classification of all those situations into like one particular limit that posits the existence of the maximal thing in all those cases awesome perhaps playing close enough attention but you so you prove that every maximal filter is an ultra filter is every Ultra filter maximal uh yeah yeah I think that one is kind of simple um okay like it's a natural condition yeah if you can make it any bigger that just involves adding one of the collage sets but the the large set is already there so that if you take the intersection yeah yeah bye I've given my background I I don't object to using Zone Slimmer um I guess what I feel I feel a little more wary of it in this context because I'm often I suppose invited to think of I guess in some circumstances you're invited to think of constructions as being more accessible than they really are in some sense right so like if deep down somewhere there's hidden an application of Zorn's Lemma but you get the impression that someone higher up the chain is trying to con you into thinking that actually you have some handle on this object and it's not just you know some yeah because like we don't even have like use magic wand you know yeah like there's no sort of thing that we can use to sort of uh access like the the maximal filter it's just it's just the kind of belief that it's there like I I guess the only in terms of like existence like a priority you might think all right you can satisfy the first four axioms um so just the filter axioms but then maybe like uh in some way trying to add the the fifth Axiom like create some internal inconsistency that like because it's not just making finally it's not just making infinitely many

arbitrary choices like at the end of it you have to sort of still satisfy the original axioms um and so you might worry that like something goes wrong in getting there hmm but I I guess because I proved it I don't know I in in applying Zone's Lemma you I guess you kind of uh you reduce it to just the task of making a bunch of arbitrary choices um yeah I'm kind of interested why why you have a sort of double I guess standard for using like axing of choice and zon's Lemma like in sort of standard mathematics and then like in this context like you're wary of it like what's the what's the what informs that I think uh I think people are just scandalously uh careless about mixing mathematics and meta mathematics and being clear about when they're doing it um and therefore I I want to raise up a red flag when it comes to reasoning I mean you're doing it but I'm saying that clearly people do it right and therefore my protest is to whinge every time somebody you know um well I mean I I think it's I think it's odd because I don't think it's the case that like okay maybe there are people that are careless and just not thinking about it but I think it's a pretty like uh it's kind of a off like a commonly held View that like metamatic mathematics is mathematics like and sort of intentionally having that perspective um like I mean you might sort of believe there is a difference and like uh like you can't I don't know maybe certain things are illegal to do in mathematic meta mathematics but I've kind of find it weird because like that just leaves metamathamatics as like a as the realm of like philosophy or something and as a mathematician it's like what I'm not allowed to use my tools to like understand this no I wouldn't put it that way I would say that okay you could I would say the default mathematical philosophical orientation is to view mathematical objects as having some sort of platonic existence and then the symbols we write down and the proofs we you know that logic is sort of about reasoning correctly about those externally existing objects and we sort of believe in them independent of any particular formalization or foundations and that I would say it's pretty clear from the way we Orient ourselves towards foundations and their paradoxes and and the way we think about you know zfc as being One Foundation but we don't we don't really believe that if we were to switch to nbg or the topos theory or whatever that like the true nature of the circle would change you know we

think we might be able to prove different statements or there might be a different flavor but we're not convinced at least absent very strong evidence that it's really different so that seems to me to indicate a pretty strong a priority belief in in something like the independent existence of mathematical objects at least a certain kind of mathematical object but what kind so okay if you take that as a given then the question becomes well how good is that logical structure that symbolic structure that we use to reach out and try and touch that platonic Realm and that's you know what I think of as uh what people like girdle were trying to do but then you see you can't really if that's how you think about what's going on that you've got the platonic realm you can't access and then you've got this puny symbolic stuff that you get to do on paper and you're asking the question well how how powerful how much can I trust this stuff then you don't get necessarily to naively think about it in the platonic way because you're you're very much engaged exactly in the question of what is the gap between the symbolic stuff and the platonic Transcendent stuff so to engage in the exercise in a spirit of well my symbolic stuff is really just the platonic Transcendence stuff is sort of to miss the point or to uh to beg the question in a way you see what I'm saying right I yeah I definitely don't think that the same thing I I guess that what's interesting to me is whether you uh what like whether the Journey of Foundations is like you were talking about the platonic realm and then the sort of symbol realm whether you start with the symbol realm and try and get to the platonic realm or whether you're starting with the platonic realm and like uh sort of attaching that to symbols I mean I feel like it's a settled question that we've like gone over many times that if you can't start with the symbol realm and then like achieve the platonic realm because various things get in the way like it goes in completeness theorem um and so I think I'm more of the mind of like uh like I like to think of uh like so somehow where like internally opinionated on this platonic Realm and uh we have some sort of internal notion of truth that we can sort of say this set is not equal to that set or that it contains this or like we have some sort of girl called it a sense very much like our senses of the physical world we strongly believes that we we have a sense of that here yeah so I I guess I'm sort of therefore

then all right can we uh start there and amongst those sort of that sense and I know the sort of mathematical sense in that uh you can can you make like like a small part of that is like the primitive tools that you need to make uh like the your your sort of logical system with all your symbols um then you then reason about that logical system um comes to conclusions uh and then sort of reflect on what that means about like the the the platonic Realm yeah I'm fine with it and that's how I would more or less think about it uh I suppose however that it's possible to be unsophisticated about that and to uh I mean that's a fairly complicated proposition that you just laid out right it's quite subtle um and I think that sometimes foundational discussions are used to try and convince people of a much more simplistic take which is sort of in some conceptual fashion exactly what we know isn't true right that we just erect the simple finite finitistic symbolic structure and it is an unerving guide to what isn't is not the case about the platonic Realm whereas what you're I mean you're very much in a modern sense uh giving up on that project almost completely red so you're sort of taking the attitude of a mathematician to logic you seem to be a mathematical logician and saying okay well uh let me study logic and Foundations very much in the spirit of a mathematician and not worry too much about trying to pick myself up by my bootstraps from zeros and ones laying on the table that I push around with a fork and get all the way to heaven um you're not seemingly very motivated by that project but many people interested in foundations deeply deeply are interested in that project I just like worry that it like I guess the discussions it sort of leads to in my experience I'm more like they just start to like they inherently become more vague and then they like I I don't know I just I just feel like you end up achieving less so I don't know you're just talking about you're just philosophizing right but maybe you should admit that the kind of questions you might be interested in are nonetheless still motivated by some aspect of that Quest right so to be interested in foundations I mean if you don't view that if you're not interested in that project in any way whatsoever then why be interested in foundations are you just kind of like oh look there's there's set theory it's about as good as geometry it just has a different flavor oh neat is that it is like I guess I'm not I don't mean to say

I'm not interested in that discussion I just find it unproductive to zoom out as in like try and go towards the like figure out what the meta theory is like um I don't know start with zfc and trying to figure out like what's the meta theory of zfc and then what's the military of that thing I think it's like I I got what exactly what I'm gonna talk about tomorrow is like um like this whole idea of like I'm kind of saying all right bring your own uh foundations to the table like somehow you've somehow you're able to like can you like talk about mathematical things in a way that makes sense in your own head um and if if you want like perfectly you've got enough talk about zfc but then you so what you're bringing to the table is in particular a hyper presumption that zfc is consistent then if that is you I mean if your meta theory has zfc in it then you just come to the table prepared to believe everything is true no so therefore you're at least what you're presuming consistency of zfc that's a fair thing to bring to the table that seems like a lot to drag to the table yeah okay let me just complete the story of sorry all right say you you've brought that to the table uh within that we can reason about a formal system uh and then we can come to some interesting conclusion about all right this formal system can't uh tell the difference between this model and that model which is clearly not like the model you had in mind um and like of course it rests on the on the sort of Prior consistency you brought to the table but I mean it still seems to say something real nonetheless yeah like it I I think like I guess my idea is instead of zooming out you like zoom in for a bit like uh discover some consequences and then try and like lift that to the The Meta Theory discussion in some way okay would you agree with the following proposition then so just as say as a geometer I might prove something about some algebraic variety and then I'm aware that I'm using zfc and if zfc turns out to be inconsistent then in principle my proof may be invalid in whatever future foundations we shift to to avoid that particular problem but I don't believe that I'm pretty sure that some version of that will remain true and therefore I feel invested in its reality in some way that's beyond the foundations I can't say what that means but I'm sort of convinced to some degree of that and in a similar way you feel some truths about sets or models or logic uh even though you're engaged in the exercise with a presumption of

consistency of zfc perhaps that those truths are somehow Beyond uh those particular foundations but then you have to worry like I mean the the proofs that seem pretty close to the axioms of zfc then seem a bit riskier in that sense right like surely you can't be as convinced of the independence of the foundations uh you're not as convinced of the truth of some proposition about the 10 sphere as you are about some proposition about sets right don't you feel that there's something a bit more dangerous about buying into the Transcendent truth of sort of model theoretic statements as versus geometric statements um well on that last point I think I don't know like this is I I view model Theory and I think the like model theorists uh talk about model Theory taking place within the like in largely the same way as like algebraic geometry takes place in like some foundations like I I didn't like it being discussed as like we're talking about metamathammatical objects but oh I don't care how people talk about it this is because they feel comfortable with it doesn't mean they should feel comfortable uh I mean I think I'm expressing a pretty commonly held feeling that when people look at set theories in a discussion a very large Cardinals there's a feeling that maybe that's not real you know maybe that's some artifact of our current foundations that won't survive it's collapse in 56 years or whatever that's a I don't think maybe people talk about it very often but that's kind of one of the reasons why people were skeptical of set theory for a long time even though they might agree that the proofs don't have problems with them it's still a lingering sense that they're not real whatever that means but anyway yeah I take your point yeah just uh I mean it sounds like I'm defending the reality of of geometry against the uh the inferior subjects of model Theory or set theory or something but that's not really true because if you look closely you can I mean there are many things in Geometry that you know if you what mamatopy theory or whatever that have a it's not like you can separate these subjects right well this you're in intuition about what is and is not natural is is constantly being refuted by experience when you get to know various ways you can embed set theoretic issues into natural geometric questions or topological questions so so yeah you have to the project of figuring out what is and is not sort of likely to survive a shift from any given foundations to another