One thing that's funny about being a grad student in the intersection of model theory and category theory is navigating how to feel about large cardinals. On the one hand, most of my time is spent doing category theory, which tends to be quite casual/unaware about the subtitles of large cardinal assumptions; on the other, I'm cursed with some knowledge of set theory---enough to know that there's more to say, but not enough to have a perfectly refined take on the matter. As a result, my feelings about large cardinal assumptions (specifically Grothendieck universes) has had a lot of changes. This blog post is about my past and current feelings on the matter.
Initially, I thought that the problem of large cardinals was a matter of consistency, and so, since the consistency strength of large cardinals was higher (lower? I forget the convention on direction here) then ZFC, it was probably a best avoided practice. However, I later realized that, at least for the assumption of inaccessible cardinals (Grothendieck universes), this is really not much of a concern. In particular, there's nothing really special about the consistency strength of ZFC, and the assumption of inaccessible cardinals is rather mild in the scale of things (if we are concerned about consistency, then replacement and powerset are much better targets then inaccessible cardinals for our concerns). Indeed, the assumption of even many inaccessible cardinals is sometimes not even depicted on the large cardinal charts, and set theorist are often assuming much worse in their day to day.
So, for a period of time, I thought that assuming Grothendieck universes was basically fine, and that, although category theorists could do a better job of noting when they make such assumptions, it didn't really matter that much. But then I did my master's thesis. For some technical reasons, I needed to consider presheaves on a large category but without making large cardinal assumptions. The tool for such a task is to consider the category of small presheaves, which is the free cocompletion of a possible large category. This category is legitimate and has many of the same properties as a presheaf category, but many notable properties (such as the existence of limits!) need not hold in this category. This makes the situation very different from taking a presheaf category by assuming universes. And so the crux of the issue is this: what the category theorist uses a single cardinal assumption to solve, usually encodes several separate assumptions. In particular, the meaning of being small as a category, a small (co)limit or a member of the category of sets could be separate things. Category theorists are not completely unaware of this issue, using terms like small, large, very large, ect, as a way to differentiate between various sizes encoded in their assumptions, but because this isn't very closely accounted for, it's hard to really say what precise assumptions are necessary in some of these constructions.
It's worth making clear again, this is not a cause to expect inconsistency. There should be some set theory in which things work out, but which set theory becomes unclear. This is made worse by the fact that different Grothendieck universes often disagree about properties of smaller sets. So I was once again convinced that we just shouldn't assume universes such assumptions.
But then recently I've been learning about independence relations in model theory, which naturally give rise to the notion of a monster model. That is, a class sized model, often defined to be saturated (for some intuition, a saturated model is basically a major generalization of the notion of an algebraically closed filed; it is a model in which all types are realized). Model theory is often more convenient inside a monster model and the existence of a monster require the existence of inaccessible cardinals; yet, model theoriests rarely claim there theorems to take place outside of set theories equiconsistent with ZFC. The way the pull this off is by computing what sort of large cardinals are needed for their constructions, meaning that the assumptions for types, the monster and automorphisms of the monster are kept separate but are related through cardinal arithmetic. The reason this is useful is that, although ZFC can't prove the existence of large cardinals, it can still talk about what would happen in a large cardinal if it existed so long as its properties are well specified. Thus, if the conclusion of a theorem doesn't itself infer the existence of large cardinals, one can often deduce, implicitly, that there is a proof of a given theorem in ZFC even if the proof that was used uses large constructions.
So now my opinion is that category theory should go about large cardinal assumptions more in this way, though I reserve the right to change my opinion.
Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
✓ Live Streaming✓ Interactive Chat✓ Private Shows✓ HD Quality✓ Free Actions
Free to watch • No registration required • HD streaming
Hey! Call me Lucy. I might make an introduction blog later, but I first wanted to make a blog-post about ultrapowers.
Ultrafilters are a concept from set theory, I'll try my best to explain what they are and why they're defined as they are.
First, a quick overview of what we will do: we will extend the real number line by adding new numbers through the use of an ultrapower, these new numbers are called "hyperreals". Roughly, this means that we will have infinite sequences [a₀,a₁,a₂,...] of real numbers representing hyperreal numbers, where similar sequences are regarded as equal. We will also show a surprising theorem: although there are seemingly more hyperreals than reals, hyperreals look the same as real numbers "from within".
If we have a sequence of reals like this: [0,1,1/2,1/3,1/4,...] (I'll call this sequence "ε"), the hyperreal that it represents can be viewed as the "limit" of the sequence. Since a large number of entries of this sequence is smaller than any positive real number r > 0, ε will be smaller than any positive real number r, but since also a large number of entries is larger than 0, ε will be larger than 0. ε is thus an infinitesimal hyperreal number. This is mostly just intuition though, so don't worry if you don't entirely get it.
Two hyperreal numbers x = [x₀,x₁,x₂,...] and y = [y₀,y₁,y₂,...] are equal if x_i = y_i for a large number of indices i. But what does "large" mean in this context?
Well, that's where the ultrafilter comes in. Ultrafilters split a family of sets into sets that are "large" and sets that are "small". In this case, we split sets of natural numbers (numbers 0, 1, 2, 3, etc) into large sets and small sets, so we have an ultrafilter on ℕ, the set of natural numbers. Ultrafilters are identified by the family of large sets: if some set A is in an ultrafilter U, then it is large, and if it's not, then it is small.
We do want our notion of "large sets" and "small sets" to make sense: for example, a hyperreal should always be equal to itself, so we want the whole set of natural numbers, {0, 1, 2, 3, 4, ...} (which is the set of indices for which a sequence is equal to itself), to be large.
Obviously, it would make sense that if a set A is large and B is larger than A, then B is also large. Thus, if A ∈ U is a member of an ultrafilter U ("∈" is the membership symbol), and if B ⊃ A contains everything A contains too ("⊃" is the superset symbol), then B ∈ U is a member of the ultrafilter as well.
We also want hyperreal equality to be transitive, thus if [x₀,x₁,x₂,...] = [y₀,y₁,y₂,...] and [y₀,y₁,y₂,...] = [z₀,z₁,z₂,...], then we want [x₀,x₁,x₂,...] = [z₀,z₁,z₂,...]. If A = {i ∈ ℕ | x_i = y_i} is the set of points at which x and y are equal and B = {i ∈ ℕ | y_i = z_i} is the set of points at which y and z are equal, then C = {i ∈ ℕ | x_i = z_i}, the set of points at which x and z are equal, includes the set A ∩ B = {i ∈ ℕ | x_i = y_i ∧ y_i = z_i}, the set of points at which x is equal to y and y is equal to z. It thus makes sense to have our ultrafilter be closed under intersections: if two sets A and B are large, then the set of points that are both in A and in B, called the "intersection" of A and B (denoted A ∩ B), is a large set as well (and thus also in the ultrafilter).
It would also make sense that, if two hyperreal numbers are nowhere equal, then they aren't equal. So the empty set, {} = ∅, is small.
The five axioms above describe a filter:
A filter F on κ is a family of subsets of κ.
A filter F on κ must contain the whole set κ.
A filter F on κ must be upwards closed, thus for every large set A ∈ F, and every larger set B ⊃ A, B ∈ F is large as well.
A filter F on κ must be downwards directed, thus for every large set A ∈ F and every large set B ∈ F, the intersection of A and B, A ∩ B ∈ F, is large as well.
A filter F on κ may not contain the empty set.
However, these are the axioms of a filter, and not of an ultrafilter. Ultrafilters have one additional axiom.
Suppose we have the hyperreal [0,1,0,1,0,1,...]: an alternating sequence of 0's and 1's. Is this equal to 0 = [0,0,0,0,...], or to 1 = [1,1,1,1,...], or is it its own thing? (Note: the 0 in 0 = [0,0,0,0,...] is a hyperreal and the 0's in 0 = [0,0,0,0,...] are real numbers, so they're different (kind of) numbers both called "0"). If it is its own thing, then is it smaller than 1? If it is smaller than 1, then it must be smaller on a large set of indices, meaning it's equal to 0 on a large set of indices, meaning it's equal to 0. If it's not smaller than 1, well, it can't be larger, so it'd only make sense if it's equal to 1, but no axiom about filters says it should! That's why we have this last axiom for ultrafilters, which makes them "decisive": for every set A, it is either large (thus, A ∈ U), or small, meaning that its complement, Ac = {i | i ∉ A}, the set of all points that aren't in A, is large.
And so we have our six axioms of an ultrafilter:
An ultrafilter U on κ is a family of subsets of κ, these subsets are called "large sets".
κ is large.
U is upwards closed.
U is downwards directed.
∅ is not large.
For every set A ⊂ κ, either A ∈ U or Ac ∈ U.
But we're still missing one thing. We can take our ultrafilter U to be the set of all sets of natural numbers that contain 6. ℕ is large, as it contains 6. It is upwards closed: if A contains 6 and B contains everything that A contains and more, then B also contains 6. U is downwards directed: if both A and B contain 6, then the set of all points that are in both A and B still contains 6. The empty set does not contain 6, and every set either does contain 6 or does not contain 6. With this ultrafilter, two hyperreals x and y are equal simply when x₆ and y₆ are equal, so we don't get cool infinitesimals and infinities, and that makes me sad :(
These kinds of boring ultrafilters are called principal ultrafilters. Formally, a principal filter on κ is a filter F on κ for which there is some set X ⊂ κ so that any set A ⊂ κ is large only if it contains everything in X. This filter is often denoted as ↑X. If you want a principal filter U to be an ultrafilter, X needs to be a singleton set, meaning it only contains a single point x. Proving this is left as an exercise for the reader.
Let U be a non-principal ultrafilter on ℕ. This post is getting a bit long, so I won't show why such an ultrafilter exists. Now, we can take the ultrapower of ℝ, the set of real numbers, by U. This ultrapower is often denoted as ℝ^ℕ/U. Members of this ultrapower are (equivalence classes of) functions from ℕ to ℝ, meaning that they send natural numbers/indices to real numbers (the sequence [x₀,x₁,x₂,...] maps the natural number i to the real number x_i). These functions/sequences/equivalence classes are called hyperreal numbers. Two hyperreal numbers, x and y, are equal if {i ∈ ℕ | x(i) = y(i)}, the set of points at which they are equal, is large (i.e. a member of U). We can also define hyperreal comparison and arithmetic operations: x < y if {i | x(i) < y(i)} is large, (x + y)(i) = x(i) + y(i) and (x · y)(i) = x(i) · y(i). Every real number r also has a corresponding hyperreal j(r), which is simply [r,r,r,r,...] (i.e. j(r)(i) = r for all i).
In general, if M is some structure, κ is some set and U is some ultrafilter on κ, then we can take the ultrapower M^κ/U, which is the set of equivalence classes of functions from κ to M, where any relation R in M (for example, "<" in ℝ) is interpreted in M^κ/U as "R(x₁,...,xₙ) if and only if {i ∈ κ | R(x₁(i),...,xₙ(i))} ∈ U is large" and any function f in M (for example, addition in ℝ) is interpreted in M^κ/U as "f(x₁,...,xₙ)(i) = f(x₁(i),...,xₙ(i)) for all i ∈ κ".
A quick note on equivalence classes: in M^κ/U, points aren't actually functions from κ to M, but rather sets of functions from κ to M that are all equal on a large set of values. Given a function f: κ → M, the equivalence classes that f is in is denoted [f]. In this way, if f and g are equal on a large set of values, then [f] and [g] are actually just equal.
The hyperreal [0,1,2,3,4,...], which sends every natural number i to the real number i, is often called ω.
This part of the blog will get a bit more technical, so be warned!
In the beginning of this blog-post, I mentioned that hyperreals look the same as real numbers. I'll make this statement more formal:
For any formula φ that can be built up in the following way:
φ ≡ "x = y" for expressions x and y (expressions are variables and "a + b" and "a · b" for other expressions a and b)
φ ≡ "x < y" for expressions x and y
φ ≡ "ψ ∧ ξ" (ψ and ξ are both true) for formulas ψ and ξ
φ ≡ "ψ ∨ ξ" (ψ or ξ is true (or both)) for formulas ψ and ξ
φ ≡ "¬ψ" (ψ is not true) for an formula ψ
φ ≡ "∃x ψ(x)" (there exists a value for x for which ψ is true) for a variable x and an formula ψ
φ ≡ "∀x ψ(x)" (for all values of x, ψ is true) for a variable x and an formula ψ
We have that ℝ ⊧ φ (φ is true when evaluating equality, comparison and expressions from within ℝ, where variables can have real number values) if and only if ℝ^ℕ/U ⊧ φ (φ is true when evaluating equality, comparison and expressions from within ℝ^ℕ/U, where variables can have hyperreal number values).
In other words: ℝ and ℝ^ℕ/U are elementary equivalent.
So, how will we prove this? Well, we will use induction: "if something being true for all m < n implies it being true for n itself, then it must be true for all n (where m and n are natural numbers)". Specifically, we will use induction on the length of formulas: we will show that, if the above statement holds for all formulas ψ shorter than φ, then it must also hold for φ.
However, we won't use the exact statement above. Instead, we will use the following:
Given a formula φ(...) and hyperreal numbers x₁,...,xₖ, ℝ^ℕ/U ⊧ φ(x₁,...,xₖ) if and only if {i | ℝ ⊧ φ(x₁(i),...,xₖ(i))} is large.
Now, why does this imply the original statement? Well, when k = 0, {i | ℝ ⊧ φ} can only be ∅ or ℕ. It being ∅ is equivalent to φ being false in ℝ and, if the statement is true, also equivalent to φ being false in ℝ^ℕ/U. And it being ℕ is equivalent to φ being true in ℝ and, again, if the statement is true, it is also equivalent to φ being true in ℝ^ℕ/U. We thus have that φ being true in ℝ is equivalent to φ being true in ℝ^ℕ/U.
Note: M ⊧ φ simply means that the formula φ is true when interpreted in M.
Now, why do we need this stronger statement? Well, it makes induction a lot easier: given that this statement holds for all ψ shorter than φ, it's easier to prove it also holds for φ.
Now, we can actually do the induction.
First, if φ ≡ "x = y", then we need to show that (1) ℝ^ℕ/U ⊧ φ(x,y) iff (2) {i | ℝ ⊧ φ(x(i),y(i))} is large. This follows immediately from the definition of equality in ℝ^ℕ/U, the same holds for "<".
Now, if φ(x₁,...,xₖ) ≡ "ψ(x₁,...,xₖ) ∧ ξ(x₁,...,xₖ)", we have that {i | ℝ ⊧ φ(x₁(i),...,xₖ(i))} = {i | ℝ ⊧ ψ(x₁(i),...,xₖ(i)) ∧ ℝ ⊧ ξ(x₁(i),...,xₖ(i))} = {i | ℝ ⊧ ψ(x₁(i),...,xₖ(i))} ∩ {i | ℝ ⊧ ξ(x₁(i),...,xₖ(i))}. Since {i | ℝ ⊧ ψ(x₁(i),...,xₖ(i))} is large iff ψ(x₁,...,xₖ) is true in ℝ^ℕ/U, and {i | ℝ ⊧ ξ(x₁(i),...,xₖ(i))} iff ξ(x₁,...,xₖ) is true in ℝ^ℕ/U, and U is closed under intersections, we have that {i | ℝ ⊧ φ(x₁(i),...,xₖ(i))} is large iff φ holds in ℝ^ℕ/U. A similar argument works for ∨.
If φ(x₁,...,xₖ) ≡ "¬ψ(x₁,...,xₖ)", then we can just use the ultraness of the ultrafilter.
If φ ≡ "∃y ψ(y,x₁,...,xₖ)", then {i | ℝ ⊧ φ(x₁(i),...,xₖ(i))} = {i | ℝ ⊧ ∃y ψ(y,x₁(i),...,xₖ(i))} = {i | ∃y ∈ ℝ. ℝ ⊧ ψ(y,x₁(i),...,xₖ(i))} = ∪_{y ∈ ℝ} {i | ℝ ⊧ ψ(y,x₁(i),...,xₖ(i))}. We have that the set {i | ℝ ⊧ ψ(y,x₁(i),...,xₖ(i))} for y ∈ ℝ is large iff ℝ^ℕ/U ⊧ ψ(j(y),x₁,...,xₖ). If this set is large for some y ∈ ℝ, and thus if ℝ^ℕ/U ⊧ φ(x₁,...,xₖ), then ∪_{y ∈ ℝ} {i | ℝ ⊧ ψ(y,x₁(i),...,xₖ(i))} is larger than that set, so it is large as well. For the converse direction, if ∪_{y ∈ ℝ} {i | ℝ ⊧ ψ(y,x₁(i),...,xₖ(i))} is large, then we can create a hyperreal z where ψ ⊧ ψ(z(i),x₁(i),...,xₖ(i)) for all i for which ℝ ⊧ ∃y ψ(y,x₁(i),...,xₖ(i)), and we have ℝ^ℕ/U ⊧ ψ(z,x₁(i),...,xₖ(i)), and thus ℝ^ℕ/U ⊧ φ(x₁(i),...,xₖ(i)). Again, a similar argument works for ∀.
(Sorry if you couldn't follow along, I'm not good at explaining these things in an intuitive way.)
This result can be extended to show that M^κ/U is elementary equivalent to M for every structure M, every set κ and every ultrafilter U on κ.
Now, this result might be surprising, as we have a new number ω in ℝ^ℕ/U. Surely, there is a formula that states the existence of this number, right?
Well, it turns out, such a formula does not exist! You can try something like "there is no natural number n so that 1+...+1 w/ n 1's is greater than ω", but ω+1 is a natural number in the hyperreals, so such a natural number does exist. Similarly, any formula you can come up with, as long as it is created using the rules above (using conjunction, disjunction, negation, qauntification, etc), cannot state the existence of an infinite number ω.
But if ℝ^ℕ/U and ℝ are seemingly indistinguishable, might there already be an undetectable infinite real number in ℝ? Well, maybe~ :3 But it's undetectable anyways, so you don't have to worry about it.
Before I end this blog-post, I want to give some more intuition on what filters & ultrafilters actually are. To me, ultrafilters, and filters in general, are like "limits of sets". The principal filter ↑X has X as limit, while non-principal filters and ultrafilters have limits that aren't really sets, but look like ones. For example, you might have the set of prime numbers in your filter, and then the limit of that filter will be a "set" in which all numbers are prime numbers. And if your ultrafilter is non-principal (so for every n, there is a set A ∈ U in the filter that does not contain n), then the limit of that ultrafilter will be a "set" in which all numbers don't actually exist. In the case of filters, this "set" can be any "set" (though it still isn't really a set), but in the case of ultrafilters, this limit looks like a singleton set (i.e. it only has one "element": ω).
I don't know if my intuition of filters and ultrafilters will help anyone, tho, but I think it's cool!
I know how Sisyphos felt because every day i open Overleaf and write about a page of my thesis but every time during writing I come up with new things that will cover two more pages. This never ends.
Proposal: Given a signature L equip the category of L-structures with some notion of 2-morphism such that the 1-equivalences become exactly the elementary L-embeddings?
A language is simply all of the extra symbols we are going to use laid out, so that when we start doing things we know these symbols are 'reserved' and we know how they are going to be read, because we affix to them an arity, that is, how many elements they take in. For example, f:1 is a function that takes in 1 element. c:0 is a constant, because it doesn't take in anything. Besides this, we distinguish between 'functions' (things that will return other elements when we assign meaning) and 'relations', things that return true or false values.
Right now, they do not have any meaning beyond that.
Formally, a first-order language 𝔏 consists of:
• A collection ℱ of function symbols, each with a fixed arity
• A collection ℛ of relation (or predicate) symbols, also with fixed arities
These are just symbols — syntax. They don’t do anything until we interpret them inside a structure.
Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
✓ Live Streaming✓ Interactive Chat✓ Private Shows✓ HD Quality✓ Free Actions
Free to watch • No registration required • HD streaming
“finite“ field field extensions with characteristic a nonstandard prime, and the inverse Galois problem
So, the inverse Galois problem: Given a (finite) group, is it the Galois group of some field extension? Or, rather, is there, for every finite group, some field extension which has that as its Galois group?
To quote Wikipedia : “ This problem, first posed in the early 19th century,is unsolved. “
Here is some reasoning that I think makes sense. It does not solve the problem (duh. This is a blog post, not a paper.) but I think it is interesting in relation to the problem.
If G is some group, and if it cannot be proven in (1st order?) Peano arithmetic that G is not the Galois group of any field extension of finite fields, or just of fields of nonzero characteristic (assuming that this statement can be expressed in Peano arithmetic in an appropriate way),
then, by the model existence theorem, there is a model of arithmetic in which there is such a field extension.
Now, I don’t exactly “know” model theory, so this next thing I say could be quite wrong, but my impression is that we can have a nonstandard model of set theory where the set whose existence is guaranteed by the axiom of infinity (i.e. our set of natural numbers), is actually the set of elements for the nonstandard model of peano arithmetic that we took.
And, this nonstandard model of set theory is living inside our normal set theory, and its sets are like, also actual sets, and the “is an element of” is the actual “is an element of” of our actual set theory. I think this is called an inner model (however, I’m not sure. Wikipedia seems to say that for it to be an inner model, it has to have all of its ordinals in common? which doesn’t seem to be what is the case in what I’m describing, so maybe it doesn’t count as an inner model.).
Anyway, so, we have our nonstandard model of set theory and of the natural numbers, and in this nonstandard model we have that there is a nonstandard prime p such that there is a field F of characteristic p, and a field extending it, K, such that the field extension K/F is Galois.
Now, if we look at one of these fields in this nonstandard model of set theory, from the perspective of our actual set theory, while it won’t be a field of finite characteristic, it should still be a field, just of characteristic 0 instead.
Because, the function for the multiplication, addition, negation, etc. will still all be valid functions from the perspective of the actual set theory, so the field axioms should still be satisfied. (characteristic 0 because there is no natural number n such that 1 + 1 + 1 + ... n times, = 0 ).
And, furthermore, the field extension should still be a field extension, and each of the field automorphisms from before should still be field automorphisms.
However, I think that, when looking at the field extension this way, there may be intermediate fields that aren’t there in the nonstandard model, because, essentially, the power sets don’t actually have to have all the subsets, only the ones that can be like, specified, or whatever.
Similarly, there may be additional automorphisms that don’t exist in the nonstandard model, but do “in reality”.
And, perhaps in our actual set theory, K/F isn’t even a Galois extension!
But, this is no issue!
The automorphisms that we had, that inside the nonstandard model comprised Gal(K/F) , all still work as automorphisms, in our actual set theory, and still form a group, just not necessarily Gal(K/F).
So, if we just take K/(the fixed field of that group) , we will then get a Galois extension, which has as its Galois group, that same group, which is isomorphic to G.
So, if my understanding of model theory isn’t too broken, then:
If it can’t be proven that there is no prime p such that G is (isomorphic to) the Galois group of some field extension with characteristic p,
then there is a Galois field extension of characteristic 0, the Galois group of which is (isomorphic to) G .
So, the possible “answers” to the inverse Galois problem for a finite group G are:
1) “Yes, there is such a field extension”
2) “No, it can be proven that no field extension produces that group as its Galois group.”
3) “No, and it can be proven that no field extension of positive characteristic produces such a Galois group, but while there is also no such one for characteristic 0, this cannot be proven.”
Actually, come to think of it, if it couldn’t be proven about the characteristic 0 case, then wouldn’t there also be a nonstandard model of, something, where there was one with char 0? Could one then extract that to get an actual one of char 0 as well? I suspect the answer is yet.
So, in that case, I think that this seems to kinda point at a way to show that:
If a finite group G is not the Galois group of any field extension, then this fact can be proven about G.
So, that’s nice.
There’s, ... probably a way to prove that without resorting to talking about nonstandard models. This is probably way overkill to show something that people presumably already knew.
(Or just wrong, but it seems right to me.)
