So here's a fake and cranky forcing notion that I created a while ago:
A condition p ∈ P iff p ∈ L(S) ∩ M and p is a closed subset of S.
The forcing poset (Skibidi) forces constructibility; take p < q iff p \subseteq q. (Both are sequences of elements, which is very important in ZF) Let p ∈ P and (it) assume[s] p ⊨ ṡ to be constructible, with ṡ being a sequence or embedding.
S in condition 1 is allowed to be non-ω_1.
However, you can't force V = L, because of L being the minimum trans. model containing all the ordinals, although I am not exactly sure what the exact details of the counter-argument against constructing a forcing notion that forces V = L are.
So I have heard of Descriptive Set Theory before, but unfortunately it came off as very bland and unmotivated for me up until now. Today, I found this theorem on one of the topology books that I have been reading (it was written by John L. Kelley), and came in the form of a practice problem:
Theorem 1. If f is a continuous real-valued function on X, then f^{-1}[0] is a G_δ. The set {0} is in the space of all reals.
My first proof of this theorem was of the following: The real numbers ℝ have a countable base, and from this it needs to be shown that the image of f^{-1}: ℝ -> {0} is G_δ. If f^{-1}{0} were somehow not G_δ, then there will be only be a finite intersection of open sets U_α in R covering f^{-1}[0] in the form f^{-1}[0] ⊂ {∩ U_α}. However, one can easily select other open sets (let's label it U_β for now) that will cover f^{-1}[0] ⊂ {∩ U_α} ⊂ {∩ U_β}; this can go on ad nauseum to infinity, as R has a countable base.
I also came up with another proof. For some B_1 and B_2 ∈ ℝ and every point x ∈ B_1 ∩ B_2, there exists a B_3 ∈ B such that x ∈ B_3 ⊂ B_1 ∩ B_2. For f^{-1}[0] in particular, and for f{B_1], f[B_2] ∈ B and every point f^{-1}[0] ∈ f[B_1] ∩ f[B_2], there is a f[B_3] ∈ B that is a subset of f[B_1] ∩ f[B_2] which contains f^{-1}[0] and is smaller than the original B_1's and B_2's. From this, it may be deduced that there is a {∩ B_α} of open sets covering f^{-1}[0], in which α is countable.
I suppose that this practice problem/theorem is somewhat trivial, but I found it to be interesting nonetheless, especially since it contained stuff relating to the Borel hierarchy. I think what I got out of it was that we can use G_δ or F_σ sets (or other sets within the Borel hierarchy :P) to analyze subsets of the real line, although I think I might need to do a bit more research on topology/DST first...
This post follows a discovery process I carried out a few years back, which led me to my favorite mathematical theory. Hope you enjoy!!!
In set theory, we operate under an ontology where "everything's a set". This works surprisingly well, as it turns out that every conceivable mathematical structure can be encoded using sets (provided your cardinals are large enough). Despite being a set theorist though, I always felt it leaves something to be desired.
What's a set, anyway? It's just an object representing a binary rule: some things are "in" it, others are not. In this way, each set S just encodes a binary function, obeying S(x)=1 whenever x∈S, and S(x)=0 otherwise. Functions always felt more fundamental to me, in this way. Everything's a function. What would happen if we took this viewpoint seriously? What will our theory look like?
Definition: We say S is a set if its outputs are all either 0 or 1.
Set(S) ≡ ∀x, S(x)=0 ∨ S(x)=1
x∈S ≡ S(x)≠0
Definition: We say S is a "pure set" if it admits a transitive superset T such that all the members of T are also sets.
A⊆B ≡ ∀x, x∈A⇒x∈B
Trans(T) ≡ Set(T) ∧ ∀x, x∈T ⇒ (Set(x) ∧ x⊆T)
Pure(S) ≡ ∃T, Trans(T) ∧ S⊆T
Axiom (ZFC): All the axioms of ZFC apply to the pure sets.
With just the above definitions, all we've done is mildly adjusted our ontology, but basically we still just have ZFC. We could consistently assert "all functions are sets" right now, and then we'd exactly have ZFC, but we don't want that. Even if we had some non-set objects though, we couldn't do much with them, so let's start there. We'll tiptoe outside our comfort zone by taking some hints from ZFCA, a mild extension of ZFC which can handle non-sets. We can just outright assert all of the axioms of ZFCA without much modification.
Axiom 1 (Extensionality): Two functions are equal if they agree on all inputs.
(∀x, f(x)=g(x)) ⇒ f=g
Axiom 2 (Foundation): Each nonempty set contains a member which is minimal with respect to membership.
S≠∅ ⇒ ∃m, m∈S ∧ ∀x, x∈S ⇒ x∉m
Axiom 3 (Specification): Given a predicate φ, every set S admits a subset containing all and only those elements obeying φ.
∃Z, Set(Z) ∧ ∀x, x∈Z⇔(x∈S ∧ φ(x))
Axiom 4 (Pairing): Any two objects can be placed into a set.
∀a,∀b,∃S, Set(S) ∧ a∈S ∧ b∈S
Axiom 5 (Union): Given S is a set of sets, it admits a union.
∃U, Set(U) ∧ ∀x, ∀y, (y∈x ∧ x∈S) ⇒ y∈U
Axiom 6 (Replacement): Given a relation φ which defines a function over a set S, the image of φ over S forms a set.
(∀x, x∈S ⇒ ∃!y, φ(x,y)) ⇒ ∃Y, Set(Y) ∧ ∀x, x∈S ⇒ ∃y, y∈Y ∧ φ(x,y)
Axiom 7 (Infinity): This one is the same as in ZFC actually.
Axiom 8 (Power set): Every set admits a powerset, which contains all its subsets.
Set(S) ⇒ ∃P, Set(P) ∧ ∀Z, Z⊆S ⇒ Z∈P
Axiom 9 (Choice): Blah blah blah let's get on with it!!!!
Now we have enough axioms to comfortably deal with any non-set objects. If we had any!!! It's almost as if we're intentionally wasting your time with a bunch of annoying book keeping (Rude. Book keeping is awesome). We want some actual functions which aren't just sets, but how can we do this without contradiction? In set theory, we can already talk about functions by encoding their graphs as sets of ordered pairs. Surely it would be safe to codify such things as actual functions...? But what should they do outside their domain? Well, our sets are outputting 0 for almost all inputs (on their proper class complement), so let's just do that.
Definition: Given sets X,Y, we say f:X→Y to express that f is a function from X to Y, obeying f(t)=0 for all t∉X.
(f:X→Y) ≡ ∀x, (x∉X ⇒ f(x)=0) ∧ (x∈X ⇒ f(x)∈Y)
Definition: A "map", from a set X to Y, is a set of ordered pairs such that each element in X is paired with exactly one element in Y.
X×Y = {p : ∃x, x∈X ∧ ∃y, y∈Y ∧ p=(x,y)}
Map(M,X,Y) ≡ (M ⊆ X×Y) ∧ ∀x, x∈X ⇒ ∃!y, y∈Y ∧ (x,y)∈M
Axiom 3.1 (Function Specification): Every map admits a function which implements it.
Map(M,X,Y) ⇒ ∃f, (f:X→Y) ∧ ∀x, x∈X ⇒ (x,f(x))∈M
Okay, so like, technically, this is all we need. We can do anything we want with sets, and now every map can become a function, so what more could you ask for? Well, there's still a ton of ambiguity in what this theory is even talking about. We keep mentioning "0" but who is that? It must be a function, but what's it do? And what does our overall universe actually look like? In ZFC, we use the Foundation axiom to control the structure of the universe, and this ends up creating a beautiful hierarchy of sets. In essence, Foundation asserts that each set is built from its strictly simpler members, without infinite regress. Can we do something similar for functions? Let's say each function is built from its nontrivial inputs and outputs, then maybe we can use that to define our foundation.
Axiom 2.1 (Support): Every function admits a support; a set containing every input for which the function's evaluation is nonzero.
∀f,∃S, Set(S) ∧ ∀x, f(x)≠0 ⇒ x∈S
Lemma: Each function admits an image.
∀f,∃S,∀x, f(x)∈S
Definition: We say f immediately precedes g whenever: f is in the image of g, or f is in the support of g. We say f precedes g, denoted f≼g, whenever there's a finite sequence of immediate predecessors with f at the bottom and g at the top. This relation is both reflexive (via singlet sequences) and transitive (via concatenation).
Theorem: Given any f, the collection {x : x≼f} forms a set.
∀f,∃S,∀x, x≼f ⇒ x∈S
Axiom 2.2 (Functional Foundation): Every nonempty set contains an element which is minimal under the predecessor relation.
S≠∅ ⇒ ∃m, m∈S ∧ ∀x, (x∈S ∧ x≼m) ⇒ x=m
Something interesting about axiom 2.2 is that it doesn't forbid a function being an immediate predecessor of itself. This can happen in the case where f is an atom, a function which obeys f(x)=f for some x. Although intuitively it feels like atoms would violate Foundation, there's actually no conflict, since the ≼ relation is already reflexive by definition. One might be tempted to reformulate Foundation into a stronger assertion, which forbids atoms, but that would be a grave error. Indeed, not only does this "weak" version of Foundation fail to forbid atoms, but in fact, atoms are forced!!
Lemma: There's a function Q such that Q(x)=Q for all x.
proof: Take any function f, construct the set S={x : x≼f}, then apply Foundation to find a minimal Q∈S. For all x, we have Q(x)≼Q≼f and thus Q(x)≼f so that Q(x)∈S, and now minimality of Q implies Q(x)=Q.
QED
Theorem: 0(x)=0 for all x, hence 0=∅={} is the empty set.
proof: Using the techniques of the previous lemma, find Q≼0 such that Q(x)=Q for all x. Let S=supp(Q), then construct Z={x∈S : x∉x} via Specification, and infer Z∉S. It follows that Q(Z)=0, but then Q(Z)=Q by construction of Q, hence Q=0. We now know 0(x)=Q(x)=Q=0 for all x. It follows that 0 is a set, by definition of Set, and in particular x∉0 for all x by definition of ∈, therefore 0=∅ is the empty set.
QED
Theorem: There exists U≠0 such that U(x)=U for at least one x.
proof: Take any f≠0, then let S={x : x≼f ∧ x≠0}. Now apply Foundation to find minimal U∈S, and note that since 0∉S then U≠0. We cannot have U(x)=0 for all x, since that would imply U=0 via Extensionality, so there's some x for which U(x)≠0. Since U(x)≼U then U(x)≼f, and since U(x)≠0 then U(x)∈S. We now obtain U(x)=U by minimality of U in S.
QED
So, using Foundation alone we've found that an atom must exist, a perfect atom which outputs itself on all inputs. All our other axioms were able to clarify that 0 in particular is such a thing. We've also found that there must exist at least a second atom, but we don't know much about it, nor can all that much be proved. Since atoms do violate at least the spirit of Foundation, we should minimize their number, so it's reasonable to insist that there are only these two atoms and no others. We may also notice that the constant 1 is tantamount to a free variable in our axioms, so we shall insist that 1 is our unique second atom. This is our final axiom.
Axiom 2.3 (Atoms): The only atoms are 0 and 1.
(∃x, f(x)=f) ⇒ (f=0 ∨ f=1)
Theorem: We have 1={0}.
proof: Previously, we found some nonzero U such that all x≼U are either x=0 or x=U. By axiom 2.3, necessarily U=1. Since 1(x)≼1 for all x, then either 1(x)=0 or 1(x)=1. It follows that 1 is a set. Moreover, any x∈1 will have x≼1 and thus either x=0 or x=1, hence 1 is a subset of {0,1}. By the set theoretic Foundation axiom (ZFC), we cannot have 1∈1, hence 1 is a subset of {0}. Since 0={} is empty, and 1 isn't 0, then 1 isn't empty. Therefore, 1={0}.
QED
It's beyond the scope of this post, but it can be shown that this theory, with all axioms included, is mutually interpretable with ZF(C). In particular, ZF can construct a class model of this theory of functions, and then embed its universe of sets inside the universe of functions in the expected way. This means the two theories are equiconsistent, among other things. It can also be shown that our theory of functions is second-order categorical, in that the aforementioned model constructed by ZF is unique up to isomorphism. This is largely due to axioms 2.2 and 2.3, and the result is akin to a function-theoretic analogue of Mostowski Collapse.
The above just motivates this theory, but it doesn't really explore any of its cool properties. If you want to investigate on your own, here are some definitions and theorems which I'm particularly fond of; you can try to prove the theorems if that seems fun to you!
Definition: Given any two sets X and Y, define exponentiation Y^X to be the set of all functions F:X→Y.
Theorem: For any sets X and Y, the exponent Y^X forms a set.
Theorem: Where 2={0,1}, the exponent 2^S and the powerset {Z : Z⊆S} are not merely equinumerous: they are literally the same set.
Definition: Let the ordered pair (x,y) denote the unique function p such that p(0)=x and p(1)=y and p(t)=0 otherwise.
Theorem: The exponent S^2 and the Cartesian square S×S are literally the same set.
Definition: Via transfinite recursion, let 𝓕[2] = {0,1}, and 𝓕[α+1] = 𝓕[α]^𝓕[α], and 𝓕[λ]=∪{𝓕[α] : α<λ} for limit ordinals λ.
Theorem: For all ordinals 2≤α≤β, we have 𝓕[α]⊆𝓕[β].
Theorem: 𝓥[α]⊆𝓕[α] for all ordinals α≥2, where 𝓥[α] denotes the α'th layer of the cumulative hierarchy of sets.
Theorem: Every function f admits an ordinal α for which f∈𝓕[α]
The 𝓕 defined above denotes the cumulative hierarchy of functions, the function-theoretic analogue to the Von Neumann hierarchy of sets. At the bottom we have the functions 0 and 1, which are atomic, and more complicated functions are obtained by considering all possible maps over simpler functions. It turns out that every function appears within this hierarchy.
