#ordinal arithmetic

okay so we did logic now we need actual stuff to do the logicking on. because you can't just do logic on nothing. it's like having a skeleton with no fat or muscle or skin like it's cool but it'd be way cooler if you had something to be a skeleton for

so let me introduce you to sets. let's just get all the obvious stuff out first: sets aren't lists. stop pretending that they are lists. you can have lists but sets are not lists. sets are:

unordered: doesn't matter how you sort it, it'll always be the same set. {a,b} = {b,a}

no multiplicity: doesn't matter how many times you repeat an element, that's the same as having it in there once. {a,a,a,a} = {a}

okay if we are clear on that let's actually get to notating sets properly. I'll also for now touch on some "basic sets" that we'll use for convenience to make the explanation easier. otherwise I'd have to start really abstractly and ion wanna do that:

ℕ is the set of natural numbers {0,1,2,3...}

ℤ is the set of integers {...-3,-2,-1,0,1,2,3...}

alright so we can first show sets as just their raw elements like I kinda did there. you usually enclose the raw elements of a set in braces. so like. let S = {1,2,3}. Then 1,2 and 3 are it's elements. you show this relationship with an ∈ symbol. so

1 ∈ S, 2 ∈ S, 3 ∈ S

and when it isn't you put a line through it

4 ∉ S

next, you can show them as a common property they share. you can just do this on it's own but it is heavily preferred that you do it to other sets.

{x∈ℕ|x is odd} would give you the elements of ℕ that are odd.

and lastly you can construct sets from other sets:

{x²|x∈ℕ} gives you the set of all perfect squares.

if you know python list comprehension comparsion becomes a lot easier:

S = [1,2,3,5] (S = {1,2,3,4}) S = [x for x in N if isOdd(x)] (S = {x∈ℕ|x is odd} S = [x**2 for x in N] (S = {x²|x∈ℕ}

now of course don't you ever DARE forget that sets are not lists so pretend that you put all of that trough a set() function too

anyways, just like with logic you can take all these sets and combine them in interesting ways

how combine the set

there are many ways:

union: A ∪ B is the set that has elements that are in either set. {1,2,3}∪{2,3,4} = {1,2,3,4}

intersection: A ∩ B is the set that has elements that are in both sets and nothing else. {1,2,3}∩{2,3,4} = {2,3}

subtraction: A \ B is the set that has elements from A but not from B. {1,2,3}\{2,3,4} = {1}

cartesian product: A × B is the set that has pairs of elements from A and B (in that order). {1,2,3}×{2,3,4} = {(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}

now let's look at some properties a set can have too:

subset: A ⊆ B (A is a subset of B) only when every element of A is in B. Note that A ⊆ A. If you don't want A = B, then you can use A ⊂ B.

singleton set: a set that only has one element. like {1}

empty set: the set with no elements. there is only one empty set. you show it as ∅ or {}.

note that the empty set is a subset of every set

there is also the power set which is the set of all subsets. You show it as 𝒫(x) but I'll just use P(x) and be obvious when I mean the power set. For example: P({1,2}) = {∅,{1},{2},{1,2}}

finally, let's talk about hereditary sets. these are sets whose elements are hereditary sets themselves too. essentially, you will not find anything other than a set in them.

whats up dipshits I am here to teach you ordinal arithmetic from the ground up but I'll do it accurately unlike every youtube video on it for some reason

we will begin with logic which is like the code that makes math run properly. without logic all is chaos and death and suffering.

how logic????

very good question mon ami the answer is simple.

propositions

propositions are the very basic statements not made by connecting two other statements. shit like "2 = 3" and "1 + 1 = 2" are propositions. propositions are either true (1) or false (0)

there are many ways to show true and false: T and F, 1 and 0, ⊤ and ⊥. I use the 0 and 1 notation because it's basically like binary. maybe if you code that should be familiar.

usually you show propositions as p,q,r,s,t etc.

logical connectives

now we will look at the different kinds of logical connectives. you tack on your propositions onto these to make even cooler statements called sentences

usually you show sentences with the greek letters φ (phi) and ψ (psi)

just like propositions they can also be true or false

here's the most common ones you'll see:

conjunction (AND): φ ∧ ψ is true only if both ψ and φ are true

disjunction (OR): φ ∨ ψ is true if at least one out of ψ and φ are true

negation (NOT): ¬φ reverses φ. if φ is true then ¬φ is false and vice versa

implication (IMPLIES): φ → ψ is a bit complicated but know that if φ is false it's always true and if φ is true then ψ has to be true for φ→ψ to be true

bi-implication/equivalence (IFF): φ ↔ ψ is true when φ and ψ are the same (both are true or both are false)

now let's look at implication more closely because the part where it being always true when the lefthand side is false is a bit weird. the reason you would do this is because if the first statement is false there isn't really anything that you could imply. anything can be true if you can't get past step one.

anyways just memorise these and you should be golden

quantifiers

these guys are those weird upside down letters you might see. they tell you about a whole bunch of stuff rather than just one thing

for example let's say I have a statement P and you apply it to x to mean "x is orange" and you show this as P(x). P(x) can be true or false

the existential quantifier means there is at least one thing that satisfies the condition you put at the front. like:

∃x(P(x)) means there exists an x such that x is orange. and there is. ginger people have orange hair so there clearly is an x that is orange. this is true.

the universal quantifier means everything satisfies the condition.

∀x(P(x)) means for all x, x is orange. this is false. I am not orange. so the statement is false.

oh and you can check these things for just a set a well. we'll explore sets too in the next part but for now know that for a set S, you would show "for all x in S, P(x) is true" as: ∀x∈S(P(x))

and I believe you're smart enough to understand what it means when you replace the ∀ in there with an ∃

part 11 of set theory (toc)

Semester's over, jokers. Here's the last contenty set theory post (I might get around to uploading solutions later to some problems that I thought were interesting).

I'm just going to dump some stuff from the end of the semester to wrap up for now. It's kind of a pity because (in my opinion) this stuff was by far the coolest stuff of the semester. Oh well.