Wow one of the subjects in math that’s too scary for me to get started. Now it’s given by one of the greatest mathematicians. Nothing more wonderful than this….
There is nothing to be afraid of. Set theory gets a bad reputation these days, chiefly for the reasons given in the second part of the video. Most of the people want you to think, that they are in a reputable kind of occupation and you can trust results of their work, while set theory exposes the provisional character of all our knowledge. Yet, in a moment in history like ours, there is no more important thing, than showing how humble our achievements are. We need utmost cautions in making our steps into the future. This does not mean, that we should, so to speak, breath more slowly, in order to push all the changes into the future. The opposite: it means, that we have to think intensely and carefully, sparing no reasonable effort in the task of understanding the world around us, notwithstanding our human condition, to which set theory seems to be a witness. I was waiting for this series since sets were first mentioned here, that is, for more than a year.
@@Suav58 Thank you very much for your insight sir. Maybe “scary” is kinda misleading. What I mean by it is some subjects in math give me no vision of what the theory is about and even I can follow the logic of the proofs and arguments, to the point that I can’t even judge whether I like/enjoy the subject or not. I’m just get lost in the sea of the notations and terminologies. One of such is Set theory, another will be number theory. It’s really hard to stay motivated to proceed.
Amusingly, starting from the axiom system ETCS, a ZFC set turns out to be *formally* defined as a rooted tree with no infinite branches and no symmetries :-) (more or less)
I noted some issues that caught my attention. As a logic learner, this video contain many important questions not only related with the set theory. 00:00 -> Axiom of Extensionality (In the version I saw, "a set is composed of objects, and a set itself is an object." Is there a difference?) 01:00 -> Axiom of Foundation which prevents Russell's Paradox (How?) 02:30 -> Replacement Axiom Defined via Functions (If a function is defined over sets, wouldn't a function defining a set be circular?) 04:40 -> Clear Explanation of the Axiom of Comprehension (There are some issues and considerations regarding the term "property." Is there linguistic confusion?) 06:00 -> Introduction to Ordinals through Von Neumann Construction 11:00 -> Representing Sets with Trees 11:30 -> Why Trees Representing Sets Can Have Infinite Children but Not Infinite Levels (What is the relation to Russell's paradox? Why is having infinite children not problematic?) 13:15 -> Meta-Analysis on Sets and Axioms (Every question asked was very exciting and provoking.) 18:30 -> "(Most of the) Mathematicians are formalists during the week day and platonists on the weekend" (What does this mean?)
An informal overview of Zermelo-Fraenkel Axioms: 1.Extensionality - A set is determined by its elements. 2.Foundation (Regularity) - No set is defined as the top of an infinite chain of sets containing the previous one: ...a3 ∈ a2 ∈ a1 ∈ a0 (In particular, ¬∃a(a∈a).) You can however, have a collection: a0 ∈ a1 ∈ a2 ∈ a3... 3.Pairing - A set can be made of any two given objects. This axiom is redundant in ZF, such a pair can be made using other axioms. (THEN WHY IN HELL IS IT STILL HERE?) 4.Union - {{1,2}, {2,3}} unionized is {1,2,3} 5.Infinity - There exists an infinite set. ¯\_(ツ)_/¯ 6.Powerset - Given a set, you can make a set of all its subsets. 7.Separation (restricted Comprehension) - Described later. Probably that given a set, you can make a set of all and only its elements that satisfy predicate P. 8.Replacement - The image of a function restricted to a set is also a set. (This is uncannily similar to Separation.) 9.Choice - Given a set of sets, ∃ a function that chooses 1 element from each, and makes a new set out of those elements. 3:15 The Language of Set Theory is First Order Logic (¬∨∀∃) plus the two symbols '∈' and '='. (Of course these require the (even more basic) concept of an object.) a∈b means that a is an element of b. (Mind that while a∈{a}, a ∉ {{a}}) a=b means that a has the same elements as b, at least according to Extensionality. Using this language we can formulate every other symbol we need. E.g. a⊆b = ∀x(x∈a → x∈b) In particular, we shall state the Axioms this way. 4:30 Set Theory USED TO be based on the Axiom of Comprehension - For any predicate P, you can form a set containing all and only the elements of which P holds. That ran into Russell's Paradox. Many axioms of ZF are forms of restricted Comprehension, made in order to avoid it. ( I am obviously not a fan of removing Comprehension and the ability of sets to contain themselves (because sentences can speak of themselves) so my solution, and the center point of my theory, lies in defining the Predicate. math.stackexchange.com/questions/1528002/a-class-that-contains-itself-as-an-element This^ says that classes contain only sets, so a Proper Class cannot contain itself, and "improper" classes containing themselves are the same as sets containing themselves. Of course, this all takes place in a GeneralComprehensionless theory that still allows sets to be members of themselves, and there sets' self-containment is perfectly fine, but the above post claims such sets are just useless to mathematics. Ok, I see it now, ZFC dealt with R.Paradox via the Foundation Axiom, and NBG via Classes. In NBG, for a class to be able to contain itself, it must also be a set, and therefore be an "Improper" class. The R.Paradox class would have to be a class of all Improper classes that do not contain themselves, and it itself would be a Proper class, avoiding the paradox. General Comprehension is only for Proper Classes in NBG. (Still, I dislike the notion of class, though accepting classes may in the end prove to be be smoother than accepting Exemptionism.) ) 5:30 A predicate is something of the language of FOL. A set is a well-defined collection of well-defined objects. What are objects? Maybe let us approach this problem from a different perspective, bottom-up, the Von Neumann hierarchy: At V0 (level 0) we have just our ∅. V1 = {∅} V2 = {{∅}, ∅} V3 = {{{∅},∅}, {{∅}}, {∅}, ∅} Vn+1 = Powerset(Vn) Here come the nasty ordinals, Borcherds said the Axiom of Infinity allows us to get to Vω. Vω = V0 ⋃ V1 ⋃ V2 ⋃... Vω+1 = Pow(Vω), because that's supposedly legal. Anyway, we can Von Neumann our way into building all those sets (almost all math can be conducted within V2ω) so this way we do not need a universal definition of "set". A set is just one of the Von Neumanned objects. 11:00 A set is usually depitcted as {a, b, c...} but it can also be drawn as a tree, with a node's children being members of the node set, and leaves being empty sets. Infinite sets have nodes with an infinite number of branches, but there is no infinite length of a branch, because the Axiom of Foundation rules out an infinite chain of sets containing sets. 13:30 Ways of interpretating the Axioms: 1 - The Platonic view - Somewhere out there exists the collection of all sets, and our axioma are a way to describe them. But the issue with is, is why is that collection not a set itself? It should be, since we can quantify over it, saying ∀sets e.g. Borcherds says that the collection of all sets is not a set because of Russell's Paradox, but I diaagree. Firstly, such a set could only exist in a theory without Foundation, because it must contain itself. In a Foundationless theory it still does not exist, because it would have to have the greatest cardinality of all sets, and we could just create its powerset, which would inherently have a greater cardinality. 2 - The Von Neumann Hierarchy - If you go out far enough and reach a Vα such that α is an Inaccessible Cardinal. Inaccessible Cardinals partially explained in notes on Hamkins' Set Theory. 3 - Multiverse view - Saying that there is no one correct model of sets, but many, none better than the others. The opposite of Platonism. To this view, Borcherds poses the question "Why do any models of set theory exist?" Platonism justifies the propriety of its theory, and so does Neumannism, but with the Multiverse it is more philosophical. (And my philosophical answer is that we need to create this model, because it describes our part of the multiverse, and we need math in here.) 4 - Formalism - Ignoring the meaning of the axioms, and just operating on them, seeing where they lead. To this view, Borcherds poses the question "Why explore this exact set of axioms then, and not some else?" (I basically answered that question at the Multiverse.) 5 - Pragmatism - Basically the same as above. 6 - Finitism - The Axioms are nonsense, infinite sets do not exist, do not care for the consequences of ZFC because it does not describe anything anyway. The problems with Finitism is that ZFC works extremely well and consistently for mathematics, infinities describe a lot of theorems.
Fantastic video! I've been trying to get into set theory for a while and this video was the perfect starting point into learning it Explained really well and in a rather entertaining way, too. Thank you so much for this video!
In the summary of positions, you left out the most significant one, which is computationalism. The absolutely meaningful statements are the computationally observable ones, in the sense that they are statements about computer program behavior, and idealizations of such to various oracles which are defined computationally by iterating the process of Turing jump. In this point of view, the reason set theory is required is because you need to define a sequence of ordinals going up in size toward the Church Kleene ordinal, and the axioms of set theory are successful precisely because they make countable models of very large height, meaning that if you generate a model of set theory by a computational theorem prover, and then pin the ordinals you produce down, meaning you order the ordinals as you prove their positions along Kleene's O, then you eventually produce a computable ordinal which is significantly larger than the one produced by other systems. The reason set theory is controversial is simply the axiom of powerset, which is an attempt to treat function types as uniform with discrete types, even though they are all continua. This works formally, because you can always get a discrete collection of functions inside a countable model. It is completely intuitively preposterous, because we have intuition for the continuity of real numbers, and the freedom to adjust real numbers to be 'totally random', which comes into conflict with the idea of an ordinal enumeration of these. In my personal opinion, this is the complete resolution to the philosophical issues. I no longer get confused about any foundation questions. That doesn't mean other people agree.
I would be interested in a comparison of this position with the position implicit in homotopy type theory, which also links notions of foundation to computational theorem proving. It seems to extended the notion of dependent type theory used in the semantics of programming languages with a more spatial perspective.
The tree picture of sets doesn't seem quite adequate in light of the fact that some of the branches should be able to merge if, say, a set contains x and y, which both contain z. The axiom of foundation would seem to imply that not only should these set graphs be of finite height, but they must be acyclic. Considering the Von Neumann hierarchy, then really every set should describe a lattice of finite height but possibly infinite width, whose upper bound is the set itself, and with a unique lower bound which is the empty set.
Not sure why you object to the case of z in x and z in y. Consider the set s = {0, 1, 2} where 0 = { }, 1 = {0}, 2 = {0, 1}. Then z = 0 in 1 = x and 0 in 2 = y. The tree for s is: s / | \ 0 1 2 | / \ 0 0 1 | 0
I was thinking the same thing. I doubt that any such thing exists yet, but initial explorations in that direction might be a combination of the multiple universe view, while accepting either the VN hierarchy or formalism (or something else, like Solomon Fefferman's explicit mathematics) to handle a distinguished concrete universe of concrete categories. You get to non-concrete categories by functors, so you don't need to worry about their having a "concrete" meaning. It seems that concrete categories are particularly suitable to (historically typical) applications in a fixed context, while other categories allow you to generalize and explore in a way that allows you to create (or agree to construct in a precise enough way) a larger context where different ways of applying or exploring can coexist (up to isomorphism! or equivalence of categories?). How do toposes come into this picture? Higher categories and higher toposes? Maybe David Roberts has some suggestions.
I thought there was the concept of an object. Something which can be a member of a set, but isn't a set itself. Does extensionality cover that implicitly? Also I thought there was the concept of a class for the collection of things which can't be put into a set. I.e. you can have the class of all sets. These concepts make things easier to understand.
I think you can (and some mathematicians probably do) define *everything* in terms of sets, even the natural and real numbers. In these cases extensionality certainly does cover the objects, as they are simply special sets.
Joe Tambourine said: "Something which can be a member of a set, but isn't a set itself." In ZFC, you don't want this to be possible. First, the predicate "in" takes only sets as arguments (that is, x in y is meaningful only if x and y are sets). More seriously, the axioms of ZFC apply only to the universe of sets. In particular, the quantifiers Ax and Ex are restricted only to x's that are sets. If you were to allow other objects that are not sets, then you would need more axioms to give properties that those objects possess (or, you could just leave those objects as being 'opaque' but this doesn't gain much since you can 'map' those objects to certain sets because nothing would be lost since those objects are opaque and possess no properties). You are proposing the idea of having urelements. For example, you could have an axiom system for sets with your idea of objects being members of sets. Then, you could combine this axiom system with Peano axiom system to talk about natural numbers and sets of natural numbers. This was probably the original idea, but they soon realized that this was unnecessary since one could define natural numbers in terms of sets and use the axioms of ZFC to 'prove' the axioms of Peano. As an aside, since every object in ZFC must be a set, this is the reason that ordered pairs are defined in that funny way using sets. I guess that one could define an ordered pair using just first order logic, but then you could not form a set containing order pairs (which of course you want to do, in particular to define set functions and points in geometry). By defining ordered pairs as sets, you then have use of all the axioms of ZFC to make deductions about ordered pairs and sets containing ordered pairs.
Classes would need axioms as well to be formal, so that wouldn't be ZFC. One axiomatisation of class theory is NBG. It's completely fine to talk about classes informally, but the important thing is that classes don't prove new things about sets. It's called a conservative extension. Class theory isn't more powerful. To be honest classes don't even make things much easier. You could define the class of ordinals and says x in Ord to indicate that x is an ordinal, but in pure set theory you would simply define a property ord and write that ord(x) is true. As for non set objets, your intuition is that some sets would be constituted of elementary objets ; but if these elementary objets do not contain sets , they would be the empty set by extentionality. In the absence of the axiom of fundation, we call atoms the sets that are their only element and they fit this intuition of being elementary. We can even build the same hierarchy as in the video with V0 the set of atoms inteads of the empty set. However it is of limited use as there is no property that tell apart an atom from another. (which is the key argument in using atoms to prove the consistency of the negation of the choice but this takes us quite far)
This may be bit of a scape goat but could a set be described by everything that it isn’t? That way you don’t have s recursive definition. Likely not useful for writing down the definition, but an interesting thought non the less.
Or perhaps mathematicians can create a new word to help define what s set is? Ex we gave a label for imaginary numbers. Is there something similar we can do for sets?
Or perhaps differentiate set and collection. Ie a collection. Contains zero or more elements. A set Is a collection where the following operations can be performed
5:29 P(a) = “a is in a”. Then there are no sets of the form A := { a| P(a) } In fact, If A is in A, then P(A) is true, but it says A is not in A, so it contradicts. If A is not in A, then P(A) is true, so A is in A, so it contradicts. Therefore, we cannot give true or false for statement P(A). So, there is some P such that there does not exist the set {a| P(a)}. 10:40 I do not understand what is motivation of von Neumann hierarchy.. 3:00 Axiom of Foundation is new for me. Descending chains are not allowed. But ascending chains are OK. 12:25 I am not sure how to make a set from the trees. For example, Let x, y, z, w be the sets of V_3 at 10:53, that is, x := { { phi, {phi} } }, y := { {phi} }, z := { phi }, w:= phi Then the following is a tree and I guess, the following tree represents the set {x, y. z, w} in V_5? x | y | z | w I do not understand almost all in this introduction. .
I’m very curious in how Von Neumann hierarchy works in practices as well. My guess is: If you want a set with n elements, then look at V_n which contains V_0 up to V_{n-1} and just rename V_i’s to whatever you like. In general, if you want a set with \alpha elements, then look at V_\alpha and you can identify elements in V_\alpha with whatever you like. But shouldn’t we define ordinal before doing all of these? I have no idea.
21:05 I would go even further to ask if anything in maths truly exists. Numbers don't exist. Neither do functions. The only connection they have to reality is their helpfulness in describing phenomena. Finitists cannot claim that it is the usefulness that makes these concepts worthy of examination either. After all, people thought zero did not have its use at some point either. Who's to say infinite sets or the infinity of infinite sets would not have their uses? Finitism seems, in my opinion, hypocritical unless the proponents of finitism do not engage in maths whatsoever and distance themselves from it like the plague.
![FNAF vs TF2 - Episode 2 [SFM]](http://i.ytimg.com/vi/z5b3V2-VLb0/mqdefault.jpg)
Would've never expected a series on set theory, but I am so happy for it! Thanks Professor!
Wow one of the subjects in math that’s too scary for me to get started. Now it’s given by one of the greatest mathematicians. Nothing more wonderful than this….
There is nothing to be afraid of. Set theory gets a bad reputation these days, chiefly for the reasons given in the second part of the video. Most of the people want you to think, that they are in a reputable kind of occupation and you can trust results of their work, while set theory exposes the provisional character of all our knowledge. Yet, in a moment in history like ours, there is no more important thing, than showing how humble our achievements are. We need utmost cautions in making our steps into the future. This does not mean, that we should, so to speak, breath more slowly, in order to push all the changes into the future. The opposite: it means, that we have to think intensely and carefully, sparing no reasonable effort in the task of understanding the world around us, notwithstanding our human condition, to which set theory seems to be a witness.
I was waiting for this series since sets were first mentioned here, that is, for more than a year.
@@Suav58 Thank you very much for your insight sir. Maybe “scary” is kinda misleading. What I mean by it is some subjects in math give me no vision of what the theory is about and even I can follow the logic of the proofs and arguments, to the point that I can’t even judge whether I like/enjoy the subject or not. I’m just get lost in the sea of the notations and terminologies. One of such is Set theory, another will be number theory. It’s really hard to stay motivated to proceed.
Amusingly, starting from the axiom system ETCS, a ZFC set turns out to be *formally* defined as a rooted tree with no infinite branches and no symmetries :-) (more or less)
I noted some issues that caught my attention. As a logic learner, this video contain many important questions not only related with the set theory.
00:00 -> Axiom of Extensionality (In the version I saw, "a set is composed of objects, and a set itself is an object." Is there a difference?)
01:00 -> Axiom of Foundation which prevents Russell's Paradox (How?)
02:30 -> Replacement Axiom Defined via Functions (If a function is defined over sets, wouldn't a function defining a set be circular?)
04:40 -> Clear Explanation of the Axiom of Comprehension (There are some issues and considerations regarding the term "property." Is there linguistic confusion?)
06:00 -> Introduction to Ordinals through Von Neumann Construction
11:00 -> Representing Sets with Trees
11:30 -> Why Trees Representing Sets Can Have Infinite Children but Not Infinite Levels (What is the relation to Russell's paradox? Why is having infinite children not problematic?)
13:15 -> Meta-Analysis on Sets and Axioms (Every question asked was very exciting and provoking.)
18:30 -> "(Most of the) Mathematicians are formalists during the week day and platonists on the weekend" (What does this mean?)
An informal overview of Zermelo-Fraenkel Axioms:
1.Extensionality - A set is determined by its elements.
2.Foundation (Regularity) - No set is defined as the top of an infinite chain of sets containing the previous one:
...a3 ∈ a2 ∈ a1 ∈ a0
(In particular, ¬∃a(a∈a).) You can however, have a collection:
a0 ∈ a1 ∈ a2 ∈ a3...
3.Pairing - A set can be made of any two given objects.
This axiom is redundant in ZF, such a pair can be made using other axioms. (THEN WHY IN HELL IS IT STILL HERE?)
4.Union - {{1,2}, {2,3}} unionized is {1,2,3}
5.Infinity - There exists an infinite set. ¯\_(ツ)_/¯
6.Powerset - Given a set, you can make a set of all its subsets.
7.Separation (restricted Comprehension) - Described later. Probably that given a set, you can make a set of all and only its elements that satisfy predicate P.
8.Replacement - The image of a function restricted to a set is also a set. (This is uncannily similar to Separation.)
9.Choice - Given a set of sets, ∃ a function that chooses 1 element from each, and makes a new set out of those elements.
3:15 The Language of Set Theory is First Order Logic (¬∨∀∃) plus the two symbols '∈' and '='.
(Of course these require the (even more basic) concept of an object.)
a∈b means that a is an element of b. (Mind that while a∈{a}, a ∉ {{a}})
a=b means that a has the same elements as b, at least according to Extensionality.
Using this language we can formulate every other symbol we need. E.g.
a⊆b = ∀x(x∈a → x∈b)
In particular, we shall state the Axioms this way.
4:30 Set Theory USED TO be based on the Axiom of Comprehension - For any predicate P, you can form a set containing all and only the elements of which P holds.
That ran into Russell's Paradox. Many axioms of ZF are forms of restricted Comprehension, made in order to avoid it.
(
I am obviously not a fan of removing Comprehension and the ability of sets to contain themselves (because sentences can speak of themselves) so my solution, and the center point of my theory, lies in defining the Predicate.
math.stackexchange.com/questions/1528002/a-class-that-contains-itself-as-an-element
This^ says that classes contain only sets, so a Proper Class cannot contain itself, and "improper" classes containing themselves are the same as sets containing themselves.
Of course, this all takes place in a GeneralComprehensionless theory that still allows sets to be members of themselves, and there sets' self-containment is perfectly fine, but the above post claims such sets are just useless to mathematics.
Ok, I see it now, ZFC dealt with R.Paradox via the Foundation Axiom, and NBG via Classes. In NBG, for a class to be able to contain itself, it must also be a set, and therefore be an "Improper" class. The R.Paradox class would have to be a class of all Improper classes that do not contain themselves, and it itself would be a Proper class, avoiding the paradox. General Comprehension is only for Proper Classes in NBG.
(Still, I dislike the notion of class, though accepting classes may in the end prove to be be smoother than accepting Exemptionism.)
)
5:30 A predicate is something of the language of FOL.
A set is a well-defined collection of well-defined objects.
What are objects? Maybe let us approach this problem from a different perspective, bottom-up, the Von Neumann hierarchy:
At V0 (level 0) we have just our ∅.
V1 = {∅}
V2 = {{∅}, ∅}
V3 = {{{∅},∅}, {{∅}}, {∅}, ∅}
Vn+1 = Powerset(Vn)
Here come the nasty ordinals, Borcherds said the Axiom of Infinity allows us to get to Vω.
Vω = V0 ⋃ V1 ⋃ V2 ⋃...
Vω+1 = Pow(Vω), because that's supposedly legal.
Anyway, we can Von Neumann our way into building all those sets (almost all math can be conducted within V2ω) so this way we do not need a universal definition of "set". A set is just one of the Von Neumanned objects.
11:00 A set is usually depitcted as {a, b, c...} but it can also be drawn as a tree, with a node's children being members of the node set, and leaves being empty sets.
Infinite sets have nodes with an infinite number of branches, but there is no infinite length of a branch, because the Axiom of Foundation rules out an infinite chain of sets containing sets.
13:30 Ways of interpretating the Axioms:
1 - The Platonic view - Somewhere out there exists the collection of all sets, and our axioma are a way to describe them.
But the issue with is, is why is that collection not a set itself? It should be, since we can quantify over it, saying ∀sets e.g. Borcherds says that the collection of all sets is not a set because of Russell's Paradox, but I diaagree. Firstly, such a set could only exist in a theory without Foundation, because it must contain itself. In a Foundationless theory it still does not exist, because it would have to have the greatest cardinality of all sets, and we could just create its powerset, which would inherently have a greater cardinality.
2 - The Von Neumann Hierarchy - If you go out far enough and reach a Vα such that α is an Inaccessible Cardinal. Inaccessible Cardinals partially explained in notes on Hamkins' Set Theory.
3 - Multiverse view - Saying that there is no one correct model of sets, but many, none better than the others. The opposite of Platonism. To this view, Borcherds poses the question "Why do any models of set theory exist?"
Platonism justifies the propriety of its theory, and so does Neumannism, but with the Multiverse it is more philosophical. (And my philosophical answer is that we need to create this model, because it describes our part of the multiverse, and we need math in here.)
4 - Formalism - Ignoring the meaning of the axioms, and just operating on them, seeing where they lead. To this view, Borcherds poses the question "Why explore this exact set of axioms then, and not some else?" (I basically answered that question at the Multiverse.)
5 - Pragmatism - Basically the same as above.
6 - Finitism - The Axioms are nonsense, infinite sets do not exist, do not care for the consequences of ZFC because it does not describe anything anyway.
The problems with Finitism is that ZFC works extremely well and consistently for mathematics, infinities describe a lot of theorems.
tks, professor! the comments on different views of set theory helped a lot so that i cannot express appreciation in finite words.
Thank you much for this video! I have not seen any real set theory lectures on RUclips. I am glad I can learn it!
Fantastic video! I've been trying to get into set theory for a while and this video was the perfect starting point into learning it
Explained really well and in a rather entertaining way, too. Thank you so much for this video!
This is the last place I ever expected a Marvel reference, but I love it anyway! Really exited for this series!
This topic is really cool to me for some reason
In the summary of positions, you left out the most significant one, which is computationalism. The absolutely meaningful statements are the computationally observable ones, in the sense that they are statements about computer program behavior, and idealizations of such to various oracles which are defined computationally by iterating the process of Turing jump.
In this point of view, the reason set theory is required is because you need to define a sequence of ordinals going up in size toward the Church Kleene ordinal, and the axioms of set theory are successful precisely because they make countable models of very large height, meaning that if you generate a model of set theory by a computational theorem prover, and then pin the ordinals you produce down, meaning you order the ordinals as you prove their positions along Kleene's O, then you eventually produce a computable ordinal which is significantly larger than the one produced by other systems.
The reason set theory is controversial is simply the axiom of powerset, which is an attempt to treat function types as uniform with discrete types, even though they are all continua. This works formally, because you can always get a discrete collection of functions inside a countable model. It is completely intuitively preposterous, because we have intuition for the continuity of real numbers, and the freedom to adjust real numbers to be 'totally random', which comes into conflict with the idea of an ordinal enumeration of these.
In my personal opinion, this is the complete resolution to the philosophical issues. I no longer get confused about any foundation questions. That doesn't mean other people agree.
@@theterriblepuddle1830 Also interested in that.
I would be interested in a comparison of this position with the position implicit in homotopy type theory, which also links notions of foundation to computational theorem proving. It seems to extended the notion of dependent type theory used in the semantics of programming languages with a more spatial perspective.
The tree picture of sets doesn't seem quite adequate in light of the fact that some of the branches should be able to merge if, say, a set contains x and y, which both contain z. The axiom of foundation would seem to imply that not only should these set graphs be of finite height, but they must be acyclic. Considering the Von Neumann hierarchy, then really every set should describe a lattice of finite height but possibly infinite width, whose upper bound is the set itself, and with a unique lower bound which is the empty set.
Not sure why you object to the case of z in x and z in y. Consider the set s = {0, 1, 2} where 0 = { }, 1 = {0}, 2 = {0, 1}. Then z = 0 in 1 = x and 0 in 2 = y.
The tree for s is:
s
/ | \
0 1 2
| / \
0 0 1
|
0
There are several "distinct" nodes in your tree with the same label. I feel it's more attractive to look at the thing as a DAG.
Really looking forward to this one!
thanks for this, always wondered about this sorta thing but never had the time to look into it
Thank you so much Professor!
I have now transitioned into a topos view with type theory as a foundation, this seems to correspond to the multiverse view?
Thanks Prof Borcherds...
When might u do a few videos on your favourite subject..Algebraic Number Theory??
Do you have any practice problems for this course?
any suggested books?
This is an unexpected way to count (spoiler: expected, I studied some math).
So is a powerset of acountable set in a set of the power continuum?
Thank you for your teaching. I greatly appreciate it
i love how it ends with "why this thing should exists"
Thanks Professor!
Thanks! I'm definitely not a finitist. There's far too much fun to be had with infinities. Finitism should really be termed 'Drearyism'..
You said it! I've always found finitism to be quite boring
What an absolute blessing
How about a Categorical interpretation?
I was thinking the same thing. I doubt that any such thing exists yet, but initial explorations in that direction might be a combination of the multiple universe view, while accepting either the VN hierarchy or formalism (or something else, like Solomon Fefferman's explicit mathematics) to handle a distinguished concrete universe of concrete categories. You get to non-concrete categories by functors, so you don't need to worry about their having a "concrete" meaning. It seems that concrete categories are particularly suitable to (historically typical) applications in a fixed context, while other categories allow you to generalize and explore in a way that allows you to create (or agree to construct in a precise enough way) a larger context where different ways of applying or exploring can coexist (up to isomorphism! or equivalence of categories?). How do toposes come into this picture? Higher categories and higher toposes? Maybe David Roberts has some suggestions.
Thank you.
And so castles made of sets
Melt into the sea
Eventually ...
Thank you!
I thought there was the concept of an object. Something which can be a member of a set, but isn't a set itself. Does extensionality cover that implicitly? Also I thought there was the concept of a class for the collection of things which can't be put into a set. I.e. you can have the class of all sets. These concepts make things easier to understand.
I think you can (and some mathematicians probably do) define *everything* in terms of sets, even the natural and real numbers. In these cases extensionality certainly does cover the objects, as they are simply special sets.
Joe Tambourine said: "Something which can be a member of a set, but isn't a set itself."
In ZFC, you don't want this to be possible. First, the predicate "in" takes only sets as arguments (that is, x in y is meaningful only if x and y are sets). More seriously, the axioms of ZFC apply only to the universe of sets. In particular, the quantifiers Ax and Ex are restricted only to x's that are sets. If you were to allow other objects that are not sets, then you would need more axioms to give properties that those objects possess (or, you could just leave those objects as being 'opaque' but this doesn't gain much since you can 'map' those objects to certain sets because nothing would be lost since those objects are opaque and possess no properties).
You are proposing the idea of having urelements. For example, you could have an axiom system for sets with your idea of objects being members of sets. Then, you could combine this axiom system with Peano axiom system to talk about natural numbers and sets of natural numbers. This was probably the original idea, but they soon realized that this was unnecessary since one could define natural numbers in terms of sets and use the axioms of ZFC to 'prove' the axioms of Peano.
As an aside, since every object in ZFC must be a set, this is the reason that ordered pairs are defined in that funny way using sets. I guess that one could define an ordered pair using just first order logic, but then you could not form a set containing order pairs (which of course you want to do, in particular to define set functions and points in geometry). By defining ordered pairs as sets, you then have use of all the axioms of ZFC to make deductions about ordered pairs and sets containing ordered pairs.
Classes would need axioms as well to be formal, so that wouldn't be ZFC. One axiomatisation of class theory is NBG. It's completely fine to talk about classes informally, but the important thing is that classes don't prove new things about sets. It's called a conservative extension. Class theory isn't more powerful. To be honest classes don't even make things much easier. You could define the class of ordinals and says x in Ord to indicate that x is an ordinal, but in pure set theory you would simply define a property ord and write that ord(x) is true.
As for non set objets, your intuition is that some sets would be constituted of elementary objets ; but if these elementary objets do not contain sets , they would be the empty set by extentionality.
In the absence of the axiom of fundation, we call atoms the sets that are their only element and they fit this intuition of being elementary. We can even build the same hierarchy as in the video with V0 the set of atoms inteads of the empty set. However it is of limited use as there is no property that tell apart an atom from another. (which is the key argument in using atoms to prove the consistency of the negation of the choice but this takes us quite far)
I can't help but love the fact the axioms of set theory could in the end not have any meaning at all
This may be bit of a scape goat but could a set be described by everything that it isn’t? That way you don’t have s recursive definition. Likely not useful for writing down the definition, but an interesting thought non the less.
Or perhaps mathematicians can create a new word to help define what s set is? Ex we gave a label for imaginary numbers. Is there something similar we can do for sets?
Or perhaps differentiate set and collection. Ie a collection. Contains zero or more elements. A set Is a collection where the following operations can be performed
Or lastly perhaps it’s okay for a fundamental statement to be recursive.
Some things just are.
Set theory was a weird but interesting subject
5:29 P(a) = “a is in a”.
Then there are no sets of the form A := { a| P(a) }
In fact,
If A is in A, then P(A) is true, but it says A is not in A, so it contradicts.
If A is not in A, then P(A) is true, so A is in A, so it contradicts.
Therefore, we cannot give true or false for statement P(A).
So, there is some P such that there does not exist the set {a| P(a)}.
10:40 I do not understand what is motivation of von Neumann hierarchy..
3:00 Axiom of Foundation is new for me. Descending chains are not allowed. But ascending chains are OK.
12:25 I am not sure how to make a set from the trees.
For example,
Let x, y, z, w be the sets of V_3 at 10:53, that is,
x := { { phi, {phi} } },
y := { {phi} },
z := { phi },
w:= phi
Then the following is a tree and I guess, the following tree represents the set {x, y. z, w} in V_5?
x
|
y
|
z
|
w
I do not understand almost all in this introduction.
.
I’m very curious in how Von Neumann hierarchy works in practices as well. My guess is:
If you want a set with n elements, then look at V_n which contains V_0 up to V_{n-1} and just rename V_i’s to whatever you like.
In general, if you want a set with \alpha elements, then look at V_\alpha and you can identify elements in V_\alpha with whatever you like.
But shouldn’t we define ordinal before doing all of these? I have no idea.
this is when mathematics becomes philosophy
Yayy!!
21:05 I would go even further to ask if anything in maths truly exists. Numbers don't exist. Neither do functions. The only connection they have to reality is their helpfulness in describing phenomena. Finitists cannot claim that it is the usefulness that makes these concepts worthy of examination either. After all, people thought zero did not have its use at some point either. Who's to say infinite sets or the infinity of infinite sets would not have their uses? Finitism seems, in my opinion, hypocritical unless the proponents of finitism do not engage in maths whatsoever and distance themselves from it like the plague.
yeeeeeeeeeeeeeeeee
Noice