Beepboopbeepboop

I was trying to send the same DM to over 100 people, and got shadowbanned at 5.

.txt for example doesn't taste like anything

.xml though is like a little treat, but you can't have too much

.xlsx tastes like steel

you and @what-goes-in-this-spot have to be more careful

if only you were more portable :(

I love you too!

i don't even remember the last time i talked to @arch-official :(

Anyways I'm tagging all of my mutuals, of which there are many:

@breadbut3d @codex-arcene @pigeonentity you three in particular tho

Kupo, Bread and Snail are already on here, y'all are wonderful too!

As for the tags, my lovely mutuals (and a few others): @terranceholdsapencil @groovygrumpyghoul @notcharadreemurr @niezwyklygrzyb @geographypolls @artisticautisticstuff @docsidxii @raspbel-art @killian-ob99 @glitterypin @thefurthestthingfromaangel @dontletmeeatpears31 @tardis-technician @spacecurlsandbowties @paulrobinsonshotel @sendhelpstars @frostbite-boy @nyctalus-noctula @lex144 @rassilonsleftbollock @dctrwhos @t-4-twissy @fluffylord @marvelmaniac715 @ixdthestreams @foolishlyzephyrus @ronavorona @t1me-v0rt3x @14thfourteenthdoctor @arwendeluhtiene

My tags:

@heaven-sent-hell-bent @heavensententhusiast @susansgirlfriend @wasteofdust @bowie-amia1 @icewarrior2000 @notbarbara-yetaxa @diaryoftruequotes @julieafan @missyqueenofevil @twissywho @river-song-has-left-the-library @girl-in-the-tardis @notjustawanderer @antergo @thesongandthesunset @songstressofnotionsgonewrong @fandomsarefamily1966 @claudiescreations @lionheartamelia @elioistired @tardisslayer @davidtennan-t @sunalsolove @chaoticwriter139 @bookwormchocaholic @b4si1wh0 @crunchyspaghetti @fruitloopsforlife @rassilonsleftbollock @thorgood @butterflysnowflake @raspberrydee @ravenclawbich @doctorjodie13 @doctah-whovian @2minutes2midnight @sempiternal-peculiarity @almostlikequake @moonys-chocolate28 @lavenderspence @manhattansangels @opentheghostfiles @the-fuckwizard @nadjem-mari @tardis-girl1985 @thatwasfunnypleaselaugh @yellow-yellow-jacket

Special mention to @transit-fag for being the one who got me into this gimmick mess

Hey mathblr, let me tell you about one of our favorite foundational systems for mathematics! It's designed to allow for unlimited self-reference, which is neat since self-reference is usually thought of as a big no-no in foundational systems. It turns out that it actually doesn't matter at all, because the power of self-reference is completely exhausted by the partial computable functions. The theory ends up being equivalent to Peano Arithmetic.

What are the axioms?

The theory is two-typed: the first type is for the natural numbers, and the second type is for functions between numbers. For convenience, numbers will be represented by lowercase variables, and uppercase variables represent functions. To prevent logical contradictions, we permit that some functions will fail to evaluate, so we include a non-number object ☒ called "null" for such cases. The axioms about numbers are basically what you'd expect, and we only need one axiom about functions.

The < relation is a strict total order between numbers.

Each nonempty class has a minimum: axiomatize the "min" operator with φ(n) ⇒ ∃m,(φ(m) ∧ min{k:φ(k)}=m≤n) for each predicate φ, and relatedly min{k:φ(k)}=☒ ⇔ ∀n, ¬φ(n).

Numbers exist: ∃n,n=n

There's no largest number: ∀n,∃k,n

There's no infinite number: ∀n,n=0 ∨ ∃k,n=S(k)

Every functional expression represents a function object that exists: ∃F, ∀(a,b,c), F(a,b,c)=Ψ for any function term Ψ. The term Ψ may mention F.

To clarify the fifth axiom, we define 0:=min{n : n=n}, and relatedly S(k):=min{n : k<n} is the successor function. The sixth axiom allows us to construct self-referencing functions using any "function term". Basically, a term is any expression which evaluates numerically. Formally, a "function term" is any well-formed formula generated from the following formation rules.

"n" is a term; any number variable.

"F(Θ,Φ,Ψ)" is a term, whenever Θ,Φ,Ψ are terms.

"Φ<Ψ" is a term, whenever Φ,Ψ are terms.

"min{n : Ψ}" is a term, whenever Ψ is a term.

In the third rule, we seem to be using the boolean relation < as if it were a numerical operator. To clarify this, we use the programmer convention that true=1 and false=0, hence (n<k)=1 whenever n<k is true, and otherwise it's zero. Similarly in the fourth rule, when we use the numerical function term Ψ as the argument to the "min" operator, we interpret Ψ as being false whenever it's 0, and true whenever it's positive. Formally, we can use the following definitions.

(n<k) = min{b : k=0 ∨ ((n<k ⇔ b=1) ∧ n≠☒≠k)} min{n : Ψ(n)} = min{n : 0<Ψ(n) ∧ ∀(k<n),Ψ(k)=0}

Okay, what can it do?

The formation rules on functions actually gives us a TON of versatility. For example, the "<" relation can be used to encode literally all boolean logic. Here's how you might do that.

¬x = (x<1) (x≤y) = ¬(y<x) x⇒y = (¬¬x ≤ ¬¬y) x∨y = (¬x ⇒ y) x∧y = ¬(¬x ∨ ¬y) (x=y) = ((x≤y)∧(y≤x)) [p?x:y] = min{z : (p∧(z=x))∨(¬p∧(z=y))}

That last one is the ternary conditional operator, which can be used to implement casewise definitions. If you wanna get really creative, you can implement bounded quantification as an operator, which can then be used to define the supremum/maximum operator!

∃[t<x, F(t)] = (min{t : t=x ∨ ¬F(t)}<x) ∀[t<x, F(t)] = ¬∃[t<x, ¬F(t)] sup{F(t) : t<x} = min{y : ∀[t<x, F(t)≤y]}

Of course, none of this is even taking advantage of the self-reference that our rules permit. For example, we could implement addition and multiplication using their recursive definitions, provided we define the predecessor operation first. Alternatively, we can use the supremum operator as a little shortcut.

x+y = [y ? sup{succ(x+t) : t<y} : x] x*y = sup{(x*t)+x : t<x} x^y = [y ? sup{(x^t)*x : t<y} : 1]

Using the axioms we established, basically as a simple induction, it can be proved that these operations are total and obey their ordinary recursive definitions. So, our theory is at least as strong as Peano Arithmetic. It's not hard to believe that our functions can represent any partial computable function, and it's only a little harder to prove it formally. Conversely, all our axioms are true when restricted to the domain of partial computable functions, so it's consistent that all our functions are computable. In particular, there's a straightforward way to interpret each function term as a computer program. Since PA can quantify over computable functions, our theory is exactly as strong as PA. In fact, it's basically just a definitorial extension of PA. Pretty neat, right?

Set theory jumpscare

Hey didn't you think it was weird how we never asserted the axiom of induction? We asserted wellfoundedness with the minimization operator, which is basically equivalent, but we also had to deny infinite numbers for induction to work. What if we didn't do that? What if we did the opposite? Axiom of finity unfriended, our domain of discourse is now the ordinal numbers. New axioms just dropped.

There's an infinite number: ∃w, 0≠w ∧ ∀k, S(k)≠w

Supremums: (∀(x≤a),∃y,φ(x,y)) ⇒ ∃b,∀(x≤a),∃(y≤b),φ(x,y)

Unlimited Cardinals: ∀a, ∃b, #(a)<#(b), where #(n) denotes the cardinality operation.

Each of the above axioms basically just assert the existence of larger and larger ordinal numbers, continuing the pattern set out by the third and fourth axioms from before. Similar to how the previous theory could represent all computable functions, this theory can represent all the ordinal recursive functions. These are the functions which are representable using an Ordinal Turing Machine (OTM). Conversely, it's consistent that all functions are ordinal recursive, since each function term can be interpreted as a program that's executable by an OTM. Moreover, just like how the previous theory was exactly as strong as PA, this theory is exactly as strong as ZFC.

It takes a lot of work to interpret ZFC, but basically, a set can be represented by its wellfounded and extensional membership graph. The membership graphs can, in turn, be encoded by our ordinal recursive functions. Using the Supremums axiom, it can be shown that the resulting universe of sets obeys a version of the Axiom of Replacement, which can be used to prove the Reflection Theorems, ultimately leading to the Specification Axiom. By adapting similar techniques relative to some regular cardinal, it can then be shown that every set admits a powerset. Lastly, since our functions are basically generated from infinitary computer code, they can be encoded by finite strings having ordinal numbers as symbols. Those finite strings are wellorderable, which induces a global choice function, proving the Axiom of Choice. Excluding a few loose ends, this covers all the ZFC axioms, giving the desired interpretation.