Hello, /sci/. This will be the first lecture in a series I will be giving here on Brave New Algebra. We will be starting from rudimentary set theory. This thread is meant to be a discussion place, so if anybody has any questions, feel free to ask. Shitposting is accepted but discouraged.
A set is naively a collection of things. We will start by imposing the axioms of Zermelo-Fraenkel set theory, coupled with the axiom of choice (sue me). But first, some definitions.
Two sets are equal, written A=B, if, given any element x, x is in A if and only if x is in B.
The product of two sets A and B is the set AxB:={(a,b) : a in A, b in B}. We will impose axioms to ensure that these two operations are allowed and produce sets.
A set A is a subset of another set B if, given any element x, x in A implies that x is in B. Exercise: prove that A=B if and only if A and B are subsets of one another. Exercise: show that the empty set {} is a subset of every set.
The union of two sets A and B is the set AUB:={x : x in A or x in B}.
The intersection of two sets A and B is the set A∩B:={x: x in A and x in B}.
A function f:A->B is a subset of AxB such that if (a,b) and (a,b') are both in f, then b=b'. Intuitively, a function assigns each element in its domain (A) to a unique element in its codomain (B). A function is injective if (a,b) and (a',b) in f imply that a=a'. A function is surjective if for every element b in B, there is an element (a,b) in f. Notation-wise, if (a,b) is in f, we write f(a)=b. Exercise: can there be a function that is both injective and surjective?
Now, the axioms.
1. Given sets A and B, there is a set AUB so that x is in AUB if and only if x is in A or x is in B.
2. Given any two sets A and B, there is a set C so that A and B are elements in C.
3. Given any set A, there is a set P(A):={S : S is a subset of A}.
4. Two sets are equal if they contain the same elements.
5. Every set A contains an element B such that A and B are disjoint. (That is, they have an empty intersection: they share no elements.)
6. (informally:) Given any formula p in the language of set theory defined over elements of a set X, there is a subset of X written {x : x in X and p(x)}.
7. There exists a set S such that the empty set is an element in S and if s is in S, then sU{s} is also a member of S. This axiom says that there exists an infinite set.
8. The product of a family of nonempty sets contains at least one element. (This is the axiom of choice.)
Let's discuss ordinals.
Definition. An ordinal is a set S such that every element in S is also a subset of S, and so that set membership defines a strict well-ordering on S.
Exercise. Show that the empty set is an ordinal, and show that no element is less than the empty set (it is the minimal ordinal).
With {}=0 the smallest ordinal, we can inductively define the successor of an ordinal S(k) to be the powerset of k. Ordinals form a proper class, and they will probably turn out to be useful somewhere down the line. I figured we should have them under our belt.
Time to talk about functions.
Let f:A->B be a function. For any element b in B, the fiber f^-1(b) is the set of elements in a such that f(a)=b.
Exercise. Show that a function is surjective if and only if the fiber over every element in the codomain is nonempty. Show that a function is injective if and only if the fiber over every element in the codomain is isomorphic to either the point (the set {*}) or the empty set. Characterize a bijective function (one which is both surjective and injective) in terms of fibers.
Given a function f:A->B, the image of f is the subset of B on elements with nonempty fiber. Show that a function surjects onto its image, and that an injection has image isomorphic to the domain.
Generalizing fibers, the preimage of a subset B' of B is the set of elements a in A such that f(a) is in B'. The preimage of the image of a function is the domain. The preimage of a single element is the fiber over that element. The preimage of the empty set is the empty set.
Given sets A and B, the set of functions from A to B is often of interest. We denote this set B^A (imagine the functions "dropping" from A to B). There is an evaluation function ev:Ax(B^A)->B which is defined by ev(a,f)=f(a). It takes an element in A and a function from A to B and yields the evaluation of that function at that element. Later on, we will characterize function sets using universal properties.
Given sets A and B, there are two projection functions p_A:AxB->A and p_B:AxB->B, defined by p_A(a,b)=a and p_B(a,b)=b. Projections are always surjective.
Exercise. Show that the fiber of the projection onto A (as above) is always canonically isomorphic to B, and visa versa.
Again, we will later show that a product and its projections satisfy a universal property. This will allow us to morally move the definition of a product to other settings using the language of category theory.
I think this pretty much concludes the lecture. I was thinking about talking about cardinals, but they will be easier to discuss later anyways. Size issues are really just a nuisance in category theory. Thanks guys. Any questions?
If not, then the next lecture will be tomorrow. We're talking about the basics of category theory! Come prepared.
>>8179028
>prove that A=B if and only if A and B are subsets of one another
A inside of B and B inside of A is equivalent to x in A implies x in B AND x in B implies x in A which by definition is x in A if and only if to x in B.
>Exercise: show that the empty set {} is a subset of every set.
We have the proposition x element of {} implies x element of A (arbitrary set).
x element of {} is a false statement and as we know, an implication that starts with a falsehood is always true. Therefore indeed x element of {} implies x element of A.
>Exercise: can there be a function that is both injective and surjective?
Consider the set {a} and {b} and the function {(a,b)}, by the definition it is injective and surjective.
>>8179048
>Every set A contains an element B such that A and B are disjoint.
Explain
>>8179075
>Exercise. Show that the empty set is an ordinal, and show that no element is less than the empty set (it is the minimal ordinal).
x element of {} implies x subset of {} is trivially true as there is no element in {} so we have an implication that starts with a false statement. ez pz.
Assume there exists some set A so that it is an element on the empty set and is a subset of the empty set.
But the empty set has no elements and therefore this is a contradiction and thus A does not exist.
ez pz.
>>8179156
disjoint means that they don't share any elements
>>8179156
otherwise, every element of A has an element of A. so there's an arbitrarily infinite containment chain.
for example, say x in A, y in A, y in x. then y in x in A. but since y in A, y has an element z in A. then z in y in x in A
and so on
>>8179163
I know that but if we take an element(say x) from a set(say s) won't that be present in the intersection of x and s. Give an example maybe?
>>8179109
>Exercise. Show that a function is surjective if and only if the fiber over every element in the codomain is nonempty.
Assume there exists son b in B such that the fiber of b is the empty set. That means that there is no a in A so that f(a) = b and therefore it contradicts the definition of surjective.
The other way around, if f is surjective then that means every b has an element a so that f(a) = b and that completes the 'only if' part of the proof.
>isomorphic
wew lad, where did you previously define that?
>canonically isomorphic
Holy shit nigga chill. You throwing this shit at me with no definition? God damn.
>I think this pretty much concludes the lecture
Congrats, no one learned anything from your lecture because you asked questions with terms you have not defined.
>>8179168
Say the set A is {1,2}. Say the element B is 1. Then A intersection B is {1} . similarly for 2. So how are A and B disjoint?
>>8179174
x won't have an element in the intersection.
the intersection of {x} and s is {x} but the intersection of x and s is something else.
for example take 3 := {0,1,2}
then 3n2 = {0,1}, 3n1 = {1} and 3n0={}
>>8179195
You are confusing subsets with elements: note that there is a difference between the set X and {X}.
>>8179195
the intersection of 1 and {1,2] is empty
1 is {0}, not {1}
>>8179178
Sorry: two sets are isomorphic if there is a bijection between them. There is not a real definition for canonical isomorphism in terms of the fibers, but you will see what I mean.
The fiber of p_A over a is the set {(a,b): b in B}. How would you construct a bijection between any such fiber and B?
>>8179048
>This axiom says there exists an infinite set.
Opinion dismissed
>>8179214
Wilberger pls
>>8179028
you haven't defined what sets are, or what elements are.
there's no way you can discuss elements of sets when we have no idea what you're talking about.
Instead of trying to use fancy words to describe what you're doing. Start with definitions and theorems. Fuck the context
>>8179075
>we can inductively define the successor of an ordinal S(k) to be the powerset of k
Aren't those the cardinals?
>>8179245
According to ZFC, elements are sets themselves. A set is any object in a model of ZFC. That's enough for a well-defined mathematical theory. The axioms are defined for the language with just one relation symbol (for inclusion), and any collection of objects in this language that satisfy the axioms above form a universe for ZFC. Any such object is a "set."
I'm avoiding deep technicalities because none of it will be important once we get into categories. Lawvere's ETCS is a better system for set theory I think; I just wanted to introduce the language and intuitions of set theory so that they will carry over when we discuss homotopy fibers and such.
>>8179284
Also what about limit ordinals, i.e. those which cannot be represented by successor functions?
>>8179284
Shit, you are right. I started writing a section on caridnals before switching to ordinals, and forgot to change it.
The successor of an ordinal k is the set kU{k}.
>>8179210
Suppose we have two sets A and B and define the operation set difference as A - B = {x ∈ A : x ∉ B}.
Now, given the sets X and {X}, what is X - {X}?
>>8179288
A weak limit ordinal is one which is not the successor of another ordinal and not 0.
Ordinals won't show up too much as we progress in subsequent lectures. I am not too concerned with getting into them. I can organize a more focused lecture on them if people are interested.
>>8179305
The axiom of foundation guarantees that X-{X} will be empty for all X (because X is the only element in {X} and cannot be an element of itself).
>>8179314
you mean X - {X} will be X for all X
How do I see that something is NOT a set?
I read that the class of all vector spaces or the class of all groups are too large to be sets. But I don't see why.
>>8179315
Right, I was describing {X}-X. Sorry. The same argument applies in both cases.
>>8179028
>A function f:A->B is a subset of AxB such that if (a,b) and (a,b') are both in f, then b=b'.
If I give you a set of ordered pairs f such that if (a,b) and (a,b') are both in f, then b=b', is f still a function? Or does a function have to contain knowledge of its domain and codomain? In the latter case, a function should be a triple (f, A, B).
>>8179028
>A function f:A->B is a subset of AxB such that if (a,b) and (a,b') are both in f, then b=b'.
Actually that's only a partial function because you didn't mention that for each a in A there has to be a b in B such that (a,b) is in the function.
>>8179326
A class won't be a set if it violates any of our axioms. The axiom 5 above, called the axiom of foundation, guarantees that no set can be an element of itself, and this can be used to show that the "set" of all sets, which must definitionally contain itself, is not actually a set as it violates 5.
>>8179349
Oh, right right. Thanks.
(Sorry for all of the errors guys; if you haven't noticed, I am trying to rush through foundations to get to the meat.)
>>8179048
>2. Given any two sets A and B, there is a set C so that A and B are elements in C.
You mean such that A and B are the *only* elements in C.
>>8179326
if you can't prove it's a set, it's probably not a set
we know of one object that isn't a set: the class of all cardinals. why? well, what's its cardinal?
so if you want to prove something isn't a set, make an injection from class of all cardinals
for example the class of vector spaces. for any set X we can make X a Q-vector space of basis X. for instance for X = {a,b,c} it's a vector space of dimension 3. So just take the injection that takes X to the vector space generated by X for each cardinal X (or hell, each set X) and there you go, it's too big
>>8179048
>8. The product of a family of nonempty sets contains at least one element.
You have not defined what a family is.
>>8179360
a "family" an "element" a "set" an "object", etc etc are all different names for a set
>>8179359
Ah, very nice and elegant. Thank you.
>>8179355
We can keep the looser version then just use comprehension, right?
>>8179028
> The product of two sets A and B is the set AxB:={(a,b) : a in A, b in B}.
This is not well defined since you have not defined what (a,b) means.
> The union of two sets A and B is the set AUB:={x : x in A or x in B}. The intersection of two sets A and B is the set A∩B:={x: x in A and x in B}.
These are not well defined as they stand. You have to use axiom 6 to define them properly.
>>8179365
Also, you have not defined what a product of a "family" is.
>>8179371
Not really. You will run into problems when trying to define what (a,b) means if you keep the looser version.
>>8179372
Okay, define (a,b) = { {a}, {a,b} }
>>8179379
if you use comprehension you don't
>>8179360
Come on guys.
A nonempty family of nonempty sets is a nonempty set S such that for all s in S, s is nonempty. The product [math] \Pi S = T [/math] is uniquely determined by the existence of projections p_s:T->S for every s, so that given any set T' and any set of functions q_s:T'->s, there is a unique function h:T'->T so that for all s p_s(h)=q_s. Semantically, this product is the set of tuples with entries in each of the s, and projections forget the other entries. No, I'm not going to build up tuples. This is an overview of set theory.
>>8179326
If, in assuming that that something is an element of a set, you get a contradiction, then that something is not a set. For example, if the collection of all sets is a set, then it is an element of itself and that leads to contradiction (because of the axiom of foundation), so it can't be a set.
>>8179392
>The product ΠS=T is uniquely determined by the existence of projections p_s:T->S
You have not defined what projections are and what T is exactly.
Sounds like you guys have an ample grasp on foundational set-theory. Excellent.
We're going over category theory tomorrow. Pretty much all of the special definitions in set theory are subsumed by limits and colimits in the category of sets as laid out in ETCS. These nuances are not the focus of this series, and when we get to the cooler stuff a lot of it won't matter at all.
A lot of higher categorical constructions work modulo variations in strictness of the coherence involved. We'll discuss it in the lecture on coherence laws. You'll see.
>>8179403
T is not a bound variable. The projections are just functions, uniquely characterized by the universal property I just gave.
>>8179287
That's true but without zfc, or stating propositional logic, you leave the audience wondering what's going on.
Everything else is fine though
>>8179386
Can you show me how to make it work with comprehension?
>>8179406
That characterization is not enough to define these functions. Can you show how to get these "projections" from the axioms?
>>8179404
when can we expect this lecture? how about in 15-20 min?
>>8179048
You're missing the empty set axiom. Without this axiom, you cannot construct any other sets.
>>8179115
>Size issues are really just a nuisance in category theory.
I don't see the issue. Just define you categories relative to a universe and everything should play out well.
>>8179421
I'm working on a paper right now. I didn't expect this much conversation on this thread; my plan was to give the next lecture tomorrow. If not, I can probably do it when I get home around midnight. I want to do it during a higher-traffic time, though.
>>8179423
The empty set can be handled by the axiom of comprehension, just select the empty formula.
>>8179418
It's enough to define everything up to unique isomorphism, which is all we will need as we go further. I don't know how to build up to it from set-theoretic foundations alone, and I don't care because I know that Set is complete and cocomplete.
>>8179428
That's what I plan on doing. Grothendieck universes are my preferred reconciliation for size issues.
>>8179392
>No, I'm not going to build up tuples. This is an overview of set theory.
If you're going to do an overview, do it properly. Otherwise, why even bother?
>>8179429
> The empty set can be handled by the axiom of comprehension, just select the empty formula.
No it can't because comprehension needs an existing set in order to apply it.
>>8179439
>just select the empty formula.
Also, what exactly do you mean by an "empty formula"?
>>8179429
>I don't know how to build up to it from set-theoretic foundations alone
That's fine. As long as you realize there are holes in your foundations (which you should fill up at some point).
>>8179447
>that version of infinity says a set exists
Your axiom of infinity already assumes the existence of the empty set, so you cannot use it to define the empty set without the definitions becoming circular.
>>8179439
Okay, there is an empty set. Thanks.
>>8179435
Because it doesn't matter up to equivalence when we start looking at various categories of sets. Even more sacrilegious, we'll move on to infinity categories and the necessary equivalences will be even looser. As I have already said, I want to touch on set theory because fibers and images will be important when talking about homotopy theory.
>>8179442
I meant the unique formula that no elements in the domain of discourse satisfy. I guess in logic it's just "false."
>>8179447
He's actually right, because the axiom of infinity references the empty set by saying that there is a set containing the empty set. Unless we can take that axiom as also guaranteeing the existence of {} implicitly? I wasn't careful when I wrote it all out. (I really don't care.)
>>8179448
Categorically, the semantics don't actually matter modulo unique isomorphism, but you are right of course.
>It's enough to define everything up to unique isomorphism, which is all we will need as we go further.
Unless you define what a "unique isomorphism" before-hand, this makes no sense.
> That's what I plan on doing. Grothendieck universes are my preferred reconciliation for size issues.
I think you're better off just ignoring foundations entirely. You've left too many things badly specified and undefined.
>>8179461
Yeah. It just felt wrong to. People are going to ask for the nitty gritties of anything I work with. Had to start somewhere.
Regarding unique isomorphisms, we'll define them tomorrow. They are isomorphisms that are unique in carrying over projection maps in this case. Universal properties will be handled rigorously in the language of categories.
>>8179455
>the unique formula that no elements in the domain of discourse satisfy
This is very vague. How do you know there is a unique formula and what exactly is the "domain of discourse". Do you want to start talking about models of set theory?
>>8179156
This is perhaps the trickiest axiom to understand. It's just there to ensure we don't have infinite containment [math] A \ni A_1 \ni A_2 ... [/math]
We don't want sets to have this kind of behavior.
Note this axiom prevents a set from containing itself, for instance.
>>8179481
I don't want to get into a talk about model theory. Let's stick with the axiom of the empty set and call it good.
>>8179156
While stating correct things, the other posters miss the point. The reason the that axiom is so important is that it says the [math]\in[/math]-relation is well-founded. This well-foundedness is vital for sets to form a cumulative hierarchy inductively built from nothing.
>>8179450
>Your axiom of infinity already assumes the existence of the empty set
Not really. The formal axiom starts with:
>There exists a set A such that there exists an a in A, and for every x, x is not an element of a
Which shows the empty set is an element without creating any circularity.
By the way, OP, I really like your idea of making lectures. Do you mind if I make one too for basic set theory? I know that your purpose is to rush to categories, and I think it'll be nice to teach the fundamentals for people who are new to the field.
Reminder that FOl is so pleb that people need to work with non-empty carrier. this is why plebs like to claim that ''there exists something in my set theory (aka there is a set)'' which cannot be demonstrated and they feel proud that all their constructions are not reduced to the empty set.
there are logics with empty carrier, but plebs despise them.
But what if you have an isomorphic homomorphic bijective surjection onto a mapped projection in R3?
>>8179337
let X = {1}
{X} - X = {{1}} - {1} = {{1}} = {X}
It's the same, what comes right after - is a set of elements you substract. So by doing {X} - X, you're not subtracting X from {X}, but the elements of X from {X}.
>>8180250
What if X = {X}?
>>8179326
When you have a set, you can create de group from this set with the group of permutations of its elements, so if there was a "set of all groups", you could make a group based on that set, which would have to contain said group, ie itself, impossible with foundation.
You can't necessarily say what is and what is not a set though, when using ZFC you actually have to use a meta theory from which you "borrow" some sets that go with ZFC, a model. So ZFC does not necessarily recognize as sets some things that could be considered as such.
>>8180144
Please, do lecture! I'd love that and you'd surely do better than I.
>>8180256
Contradicts foundation
>>8180256
Whoops, missed the earlier discussion abt the axiom of foundation. My bad.
You're doing the lord's work, OP.
>>8179048
Given these axioms, how does one take a union of infinitely many sets?
>>8180271
Suppose you want to take a union of all sets in S (a family of sets). Then x is in US if and only if there is at least one element of S which contains x.
>>8180271
You can't.
>>8180144
s/formal/correct
>>8180276
So, are you using comprehension or something? Given those axioms, how does one construct the set that you describe?
>>8179569
Actually, the well-foundedness if vital if extensionality is to work properly.
>>8180286
You are correct I think. Anybody know if ZFC requires the axiom of union to be for infinite unions? I thought finite would work because we have the axiom of pairing, but that simply shuffles the issue around.
>>8180258
>You can't necessarily say what is and what is not a set though, when using ZFC you actually have to use a meta theory from which you "borrow" some sets that go with ZFC, a model.
The part about "borrowing" is wrong and doesn't make much sense.
>>8180292
Yes, you require the union axiom to allow "infinite unions".
>>8179339
Yes, it seems the OP is considering functions to be triples (f, A, B). So I guess OP uses f:A->B as a shorthand for (f, A, B).
>>8179339
>>If I give you a set of ordered pairs f such that if (a,b) and (a,b') are both in f, then b=b', is f still a function?
no, your writing lacks ''for all a and b, b' ''
>>8180262
Great! I'll see when I have the time, I'll probably have something ready in a few weeks.
>>8180258
In set theory as fundations, sets are not defined, just as in category theory as fundations, categories are not defined.
in category theory as fundations, you can define a set, just like you can define a category in set theory as fundations
Why do I need to care about axiomatization of set theory? I'm a math grad student and I always used naive set theory, treating sets as a black box on which all operations simply work, and I never ran into any problems.
>>8180373
fundations ask the question ''why do math?''. of course a pleb students like you is not meant to reflect on his work...
>>8179048
>the product of a family of nonempty sets
What is a "family"? A set?
>Ordinals form a proper class
What is a "class"?
>Show that the fiber of the projection onto A (as above) is always canonically isomorphic to B, and visa versa.
What is (canonically) isomorphic?
>>8180376
>saving thumbnails
>>8179048
Im curious, isn't the second axiom kind of weird?
Why pair? I would propose a simpler and equivalent axiom
2a. For any set A, there is a set B such that A is the only element of B.
With the axiom of summation this implies 2. and is simpler. Or am i missing something?
>>8180373
>I'm a math grad student
How is this possible without at least learning about AC?
>>8180395
I just use axiom of choice (or actually only Zorn's lemma) whenever needed. Where's the problem?
>>8180376
>What is a "family"? A set?
You say family or something similiar whenever you're too lazy to check if you collection actually forms a set.
>What is a "class"?
Every set is a class, every collection too big to be a set is a proper class.
>What is (canonically) isomorphic?
Look up natural transformation on wikipedia or nLab to find out what the rigorous notion of canonical is. I guess you can do without that for a while though and just learn what a "regular" isomorphism is.
>>8180396
Alright then, I just thought you never learned about that.
>>8180397
the set theory is a fuking joke
>introduce rigorous notions about what is a set
>oops some collections are not sets now
>lets find a new word for them ("classes"), problem solved
>>8180397
>You say family or something similiar whenever you're too lazy to check if you collection actually forms a set.
Might as well be lazy from the start.
>>8180388
You can do it that way and it looks simpler at first, but you will make other things complex once you start fleshing everything out.
>>8180373
That's because you've never done any formal proofs.
>>8180420
now that i think of it:
op wrote the sum and intersection axiom in a different way than usual
if they were formulated as usually, "given a set S, the union over its elements is also a set", then you need the axiom of pair to let you sum two sets.
>>8180403
>>lets find a new word for them ("classes"), problem solved
It doesn't work that way, the use of classes in ZFC is informal.
The correct solution is to either use NBG set theory, which has a primitive distinction between sets and classes, or (the common solution) to assume we're working inside a standard universe U, that statisfies all axioms of ZFC.
>>8179028
>Brave New Algebra
thank you, Mister Huxley
>>8180427
>sum two sets.
What do you mean by "sum"?
>>8180298
What ?
Consider a meta-theory with a model of ZFC containg a smaller model of ZFC.
Now if you're using the smaller model of ZFC, the sets that are in the bigger model but not in the smaller are considered sets in the meta-theory, but not in your smaller model of ZFC.
Might have been confused but you'll sure get what I meant and will be able to correct me.
>>8180403
>not doing set theory
>does not introduce rigorous notions about what is a set
>oops this huge "set" is paradoxal and doesn't make sense
>let's do nothing, problem solved
>>8180898
>Consider a meta-theory with a model of ZFC containg a smaller model of ZFC
Can you give me an example of this? This seems kind of bogus.
>>8180930
Literally any large-cardinal axiom. Models in models in models.
>>8180934
So by "model", you mean a Grothendieck universe.
>>8180942
That's a very small large cardinal axiom which necessitates the existence of smaller models of ZFC, so serves as an example. Though it's set-theoretically not even worth mention as being "large cardinal"; it is of the same consistency strength as there existing an inaccessible above any ordinal.
Even the smallest non-trivial large cardinal axiom, Mahlo cardinals, absolutely dwarfs the requirements of the Grothendieck universe.
>>8180942
Grothendieck universes provide the examples you wanted yeah, but I meant model as in model theory.
>>8180970
Yeah, with sufficient large cardinals, one can certainly simulate the metamathematics of the model theory of set theory of the model theory of set theory; really, it's completely natural.
One can just never completely "catch up" to oneself.
>>8179206
I'm not the person you're responding to and I didn't read most of those first posts because I found it painfully informal.
That said, why are you giving such a terrible definition for isomorphism? Clearly a better one would be to say that when you have two maps
f:A->B
and
g:B->A
such that
fg:A->A is the identity on A and gf:B->B is the identity on B (using diagramatic composition) then we are said to have an isomorphism between A and B.
It is true that on sets without any structure these maps are precisely the bijections but if we are talking about more complicated structures (groups, rings, topologies, etc..) then the definition of an identity changes and thus bijections aren't necessarily isomorphisms.
>>8180152
Could you elaborate on what you mean by empty carrier?
I agree with the issue you mentioned about undefinable sets. I find set theory really interesting but I don't think it's actually capable of giving the constructions people like to assume it has. Even basic things like the real number constructions give you weird problems that most people ignore once you really get into the nitty gritty of formal definition.
>>8180440
>It doesn't work that way, the use of classes in ZFC is informal.
So much this. Seeing OP talk about classes in the context of ZFC made me puke in my mouth a little.
![normanwildberger[1].jpg](https://i.imgur.com/fcyld6m.jpg "normanwildberger[1].jpg")
>>8179028
>Set Theory
>>8180977
Until I see this done formally, it all seems bogus to me.
>>8181044
>Even basic things like the real number constructions give you weird problems that most people ignore once you really get into the nitty gritty of formal definition.
What weird problems are you talking about?
>>8180376
>What is a "family"? A set?
The term "family of sets" originates from naive set theory (you can look up a definition in Halmos' book if you like). The definition becomes somewhat meaningless in formal ZFC because everything in ZFC is a set. In general we say that a set F is a family of sets if F contains a bunch of sets that each contain some objects of interest.
For instance, suppose I made a bunch of sets that each contain multiples of a number. Like A_3 contains all multiples of 3, A_4 contains multiples of 4, and so on. In general
A_i contains all multiples of i
Then I could define a set
F={A_i | i is an integer, i>1}
In this case F is a family of sets.
>What is a "class"?
see >>8180440
Again it is an example of OP being informal. There is another version of set theory called NBG. In NBG there exist both "classes" and "sets" (contrast with ZFC which only has "sets"). There every set is a class but not every class is a set. if a class is not also a set then it is called a "proper class". this is shown through contradiction because sets aren't allowed to have certain properties (otherwise this creates paradoxes in the theory).
In NBG a proper class is typically said to be "too big" to not be a set but there are other reasons why it fails. In NBG, an example of a proper class would be the "class of all sets". In ZFC there simply does not exist a set of all sets (non-existence is the case for all NBG "proper classes" in ZFC).
>What is (canonically) isomorphic?
The definition of isomorphism is given here >>8181027 and elsewhere in the thread.
"Canonical", on the other hand, is harder to define. Roughly it is used for special and unique maps that satisfy some properties (a proper explanation requires a good amount of category theory background). It is, in my opinion, very poor writing and I've never agreed with its use.
>>8180397
literally everything in this post is wrong.
>>8181118
>For instance, suppose I made a bunch of sets that each contain multiples of a number. Like A_3 contains all multiples of 3, A_4 contains multiples of 4, and so on. In general
>A_i contains all multiples of i
>Then I could define a set
>F={A_i | i is an integer, i>1}
>In this case F is a family of sets.
So a "family" is a set obtained using the axiom of replacement?
>>8179028
wait this is tex file or something? cool...
anyways...
1. We know that A=B if and only if A and B are subsets of one another because of the following.
I will perform this proof via contradiction.
Let's suppose A and B are subsets of one another, but A=/B. We can therefore say that x in A implies that x is in B and vice versa. This is the definition of an equal set and, therefore, we find that the sets are equal-a contradiction. Thus, if A and B are subsets of one another, they must be equal.
Now let's suppose they are equal but not subsets of one another. Thus we can say that if x is in A, then x will also be in B for any element x in A. The same can be said for any element y in B. This is the definition of a subset, and thus, another contradiction arises. We can conclude that if A=B, A and B are subsets of one another.
Therefore, A=B if and only if A and B are subsets of one another.
holy shit I just wrote out different attempts to prove that like 5 times...
2. We know that the empty set is a subset of every set for the following.
The empty set is defined such that there are no elements existing within it. This proof will be conducted via contradiction.
Let's suppose the empty set is not a subset of the set A, an arbitrary set. This means that there exists some element in the empty set which is not in A. Since we know that there are no elements in the empty set, we know this cannot be true by definition. Thus we arrive at a contradiction. Therefore, the empty set is a subset of every set.
how do you define nothing??? philosopy...
3. Yes. I will provide an example.
A={1,2,3}, B={a,b,c}
f:A->B is defined such that f(1)=a, f(2)=b, f(3)=c
>a function assigns each element in its domain (A) to a unique element in its codomain (B)
this is wrong. the element is not always unique.
I support your project OP.
Can you outline where you want to get at (with "Brave New Algebra") and when?
>>8181147
>a function assigns each element in its domain (A) to a unique element in its codomain (B)
I forgot to mention that a function does not always assign an element of A to B
ty for lesson tho. very nice going over proofs and set theory
>>8181151
>I support your project OP.
Me as well OP
>>8181107
I hope you don't mind boilerplate math, because that's what it comes down to.
1. Consider that a set in ZFC exists iff you can write a definition for it in the language of set theory (raw formal logic).
2. Recall that ZFC is given as a set of axioms over a formal logic. Furthermore, recall that that formal logic is given by a formal language over a finite alphabet and "sentences" in formal logic must be of finite length.
3. In Formal language theory there is a basic theorem that tells us that the set of all "sentences" (finite strings) in a formal language over a finite alphabet is countably infinite.
4. Therefore the set of sentences in ZFC is countable and thus only a countable number of sets can be defined (i.e. exist). I will call such sets: definable sets.
5. Since the reals are uncountable, then in any construction or encoding of the reals there will be an uncountable portion which cannot actually be defined (i.e. we can define a countable number of irrational numbers such as sqrt(2) and pi but there will always be an uncountable number of reals that cannot be given a finite definition). Such reals are called definable reals.
https://en.wikipedia.org/wiki/Definable_real_number#Definability_in_models_of_ZFC
(cont.)
>>8181192
(cont.)
These put us in a really awkward position.
>Note that we can deduce properties of undefinable reals as part of a collection but never about any individual undefinable real (otherwise we could abuse the property to define the real).
>Can we define a set containing only undefinable reals? Probably not. If we could then the axiom of choice would actually give a contradiction. On another note the only way of showing such a set is not empty is by contradiction (no element could ever be picked out).
>Can we define the set of all definable reals (or equivalently the set of all undefinable reals)? No.
>If a set contains at least one undefinable real then it must contain uncountably many.
>Every uncountable set of real numbers that you have ever worked with contains a countable but undefinable "subset" (can it really be called a set) containing only definable reals. This "subset" if it existed could act as a drop-in replacement for the original set and you would never be able to tell the difference. However, such talk of a theory of definable reals is obviously crazy, not to mention even more problematic than this.
In general, these undefinable reals are like black matter that fill up most of the space inside our uncountably infinite sets. We can't pick them out, we can't describe them, we can't really say they exist on any formal level; and yet without them we can't even deal with basic things like the real numbers.
>>8181132
No, no, keep in mind that this is an informal term (from a formal perspective, every non-empty set can be regarded as a family). A family of sets can be defined in any way. The only thing is that it is a collection of other sets.
Typical uses for families of sets are defining families of maps between families of sets, proving whether or not a set is contained in a family, and invoking the axiom of choice over said family. There are many other uses as well.
There's a wiki page as well.
https://en.wikipedia.org/wiki/Family_of_sets
>>8181193
OK, I admit that is weird. I think the fundamental problem comes from the fact that ZFC is a first-order theory. Maybe we need to use a set theory based on second-order predicate logic instead.
>>8181027
In Set, they are obviously equivalent. The point was to have people derive that isos are jointly surjective (epi) and injective (mono) in Set, because I think it is valuable to show later on that this is not the case in general.
>>8181151
Expect a more thorough category theory lecture in the next few hours. I plan on going through higher groupoids, (∞,1)-categories, then talking about coherence laws, E[k]-algebras, E∞-algebras, and then having some discussions on operads.
>>8181217
I'm not saying that a second order set theory wouldn't be interesting or valuable but it would still have the same problems I listed (specifically you would still be restricted to a countably infinite number of definable sets). The problems I mentioned come directly from formal language (thus it affects all formal logics). In fact, the same problem affects natural languages as well and gives us a bunch of corollaries like:
>The set of all possible written text is countably infinite.
>The set of all theorems/proofs/theories/etc.. is countably infinite.
There are a few possible "workarounds" but I think they may actually be worse than the problems they're supposed to fix. Stuff like allowing sentences in formal logic to have infinite length (such sentences cannot be proven true or false).
Personally I think the cause for much of the weirdness in the ZFC boilerplate comes from stuff like the axiom of choice and the law of the excluded middle (i.e. positive proof by double negation). It just doesn't seem like these things are true at this level of detail.
>>8181228
Aha, so you read Lurie and want to translate it to your own terms (gaining some clarity and confusing everybody else)?
I'll read it, if you make a strong case for it (motivation) and include some computations to do.
I've still not found my personal application for [math]\infty[/math] cats, and I don't know E-algebras, but I like philosophizing about equality (I just realized that this is what Homotopy Type Theorists and feminists have in common)
>>8181239
Thankfully this "series" isn't going to be about ZFC anyway.
>>8181273
I'm just getting into Lurie; most of what I have read had come from Leinster's Higher Algebra and the nLab. However, teaching stuff helps me learn it better. (Also, that's a funny lampoon at HoTTists.)
Yeah, I didn't expect people to get so into the foundations. I just wanted to brush up on some simple set theory before moving on to categories, since set theory is just 0-category theory. I definitely should not have brought ZFC into it.
>>8181338
I was under the impression that the category set isn't really the same thing as ZFC.
>>8181359
It's not.
He means that if you consider a category and strip off all arrows, you're left with a "collection". And if you still look at what functors do, they are reduced to functions. But nobody ever takes that perceptive.
>>8181362
The foundations you choose for "discrete categories," ie for set theory, ripple all the way up the categorical levels. This is why anafunctors were invented, for example. I'm assuming ZFC for my discrete categories/sets.
>>8181369
I should concede: there definitely is a difference between ETCS (the "standard" theory characterizing "small discrete categories" and tying set theory and category theory) and ZFC. I don't know if ZFC strengthens ETCS. It won't matter much honestly as these lectures continue, though. I'm not stressing out over it, and I hope people take away the right lessons from the threads to come.
(And, to update you guys: I have typed out most of the category theory lecture. I will be busking for four or five hours, but I plan to make the post around ten or eleven.)
NICE THREAD!
>>8181428
>around ten or eleven
doesn't mean a thing
About the ZFC/ECTS thing (that doesn't matter), I'm 107% sure it's discussed to death by the nLab crew.
I'm also sure ZFC is strictly stronger. But I'm also not sure what you mean anyway.
I assume you want to have a category of sets where the objects are the sets that together are (a model of) ZFC. But that's something else than saying
>I want ZFC for my discrete categories.
If you throw away all arrows, do you have a discrete category in your theory with objects all distinct groups (a collection that would be at best a class, not a set internal to ZFC) or not? I'm sure you have a bunch of categories with all groups the objects. Then you can't have just ZFC for your discrete cats, because (if I remember the argument correctly) in a category of sets, each set can be equipped with a trivial group structure, making it into a category of groups so if you have actually merely all group objects lying around in a category, you have a copy of all sets in the category too and that's not well founded. Whateva, gn8
>>8181102
That's a natural reaction, though you'd have to study set theory a while for the recursive model-theoretic simulation to become natural. But many proofs involve statements about a model modelling that an inner model models so-and-so, which formally looks like recursively nested model-theory notation.
The reals are w_1. Algebraic structure is not the goal of set theory.
Lecture 2: Category Theory
>>8182284
Can you explain to me what the proper use of the colon is in definitions, and also := please?
>>8182406
Yes, A:=B or A=:B is just shorthand for "A is defined to be equal to B."
>>8182406
"[math] A := B [/math]" [math] := [/math] "[math]A [/math] is defined to be equal to [math]B[/math]".
Put more simply, [math] := := := [/math].
This is how I introduce the notation to my students. I just write [math] := := := [/math].
>>8183414
[math] (A\simeq B)\simeq (A=B) [/math]
??
>>8183427
>"Similar or equal to"
Very clever. Yes. [math]\simeq \simeq = [/math].
>>8183442
(He was stating Voevodsky's univalence axiom, if you are interested. Pretty neat!)
