#Axiom-1

[lonestarflight]

"Axiom-1 (Ax-1) Liftoff

A SpaceX Falcon 9 rocket, carrying the company's Crew Dragon spacecraft, lifts off from Launch Complex 39A at NASA’s Kennedy Space Center in Florida at 11:17 a.m. EST on April 8, 2022, on Axiom Mission 1 (Ax-1). Commander Michael López-Alegría of Spain and the United States, Pilot Larry Connor of the United States, and Mission Specialists Eytan Stibbe of Israel, and Mark Pathy of Canada are aboard the flight to the International Space Station. The Ax-1 mission is the first private astronaut mission to the space station."

NASA ID: KSC-20220408PH-KLS01_0031

Date: April 8, 2022

[codypunk]

fraxiom friday

[kalinara]

(Fall of the House of X #3)

This set of panels has two of my favorite things:

Cyclops showing off his ability to navigate and maneuver without sight, even in the most dire circumstances.

Cyclops being able to find common ground and make an alliance with a former enemy for the greater good.

Possibly three 3. Having enough weird sexy mojo to get a lady who hates him entirely to save his life and follow him into a sewer.

[th3-pup-3v3r]

I'm sobbing and screaming and throwing up. Where are you, Axiom's End/Noumena series by Lindsay Ellis fans...?? I NEED TO BE IN THE FANDOM BUT I CAN'T FIND IT!! Augh.

[corntort]

i animated it

[kyouka-supremacy]

---

[yunalai]

GET TO KNOW ME: TOP 10 MOVIES

━━━ list your top 10 favorite movies and tag friends to do the same

Spider-Man: Across the Spider-Verse (2023)

Spider-Man: Into the Spider-Verse (2018)

Nope (2022)

Treasure Planet (2002)

Ready or Not (2019)

Brother Bear (2003)

Finding Nemo (2003)

Devil's Rejects (2005)

Harley Quinn: Birds of Prey (2020)

A Knight's Tale (2001)

tagged by: @mxldito (stanky)

tagging: @axewhirl @noxianwill @ionianwanderer @agonizedembrace

[twohundredheroicaxioms]

One: First, do good.

[l-cereta]

Interestingly, though, (and extremely specific to this example! This is probably where op's friend's husband got it from) in a study by Haidt, Koller, and Dias in 1993 (Affect, culture, and morality, or is it wrong to eat your dog?), they interview Americans and Brazillians on this exact question — and found that despite their being no harm against others in the act, AND despite participants directly stating themselves that no one was harmed... "participants still usually said the actions were wrong, and universally wrong. They frequently made statements such as, 'It's just wrong to have sex with a chicken'" (my emphasis). Eight years later, Haidt would go on to argue in a seminal paper in the field of descriptive ethics (i.e. "What do people actually do?" as opposed to "What should people do?") that people form moral positions due to knee-jerk reactions and then justify their positions after the fact, as opposed to trying to privately reason things out and then come to a conclusion. I'm not a fan of the so-called 'Wisdom of Repugnance' any more than the next tranny, but the reason that it got spread around so much is that it's how we — for better and often for worse! — actually think.

This is a sort of different tangent than what @jurph is on (and for that conversation all I'll voice is a skepticism that moral thought, or really anything, can be broken down into neat categories as the Wikipedia article seems to suggest — seems like Haidt also had a hand in that, oh well, can't win 'em all) but I think the crucial point is that this isn't something that conservatives uniquely do. In Haidt's later 2001 paper, he mentions another study earlier that year where they "found the same thing when they interviewed conservatives and liberals about sexual morality issues" (my emphasis). Whether using a dead chicken like a fleshlight actually harms someone matters a lot less in the actual judgement-making than whether it feels wrong: "affective reactions were good predictors of judgement, whereas perceptions of harmfulness were not."

I'm very sympathetic about these "should" statements. For example, we shouldn't judge people based on unfair first impressions — but we do, and we probably always will (take, for instance, this 1972 study indicating that a person looking attractive leads other people to think they're also kind and of good character). Morals shouldn't play into our sex lives, but they do, and it's not just the conservative boogeymen who do this. Everyone does. Instead of shaming someone who's behaving in the default manner for every human on earth, talk to them like a real person! Haidt writes about a phenomenon called 'attitude alignment,' where there's "a constant pressure towards agreement if two parties [in a conversation are] friends and a constant pressure against agreement if the two parties [dislike] each other." By speaking harshly, you're lessening your odds for agreement!

Oh my god. I need to share another story of my new friend making today. So my friends husband says, very casually, as we’re about to leave for the ren faire, “Yeah, it’s like my story about fucking a chicken.”

And of the four people present I was the only one who was shocked. The others all nodded as if to say, yes yes, we know, the chicken fucking.

So he explained, when a progressive person is analyzing a behavior they will typically use the metric, Harm/No Harm. They may not like things in the No Harm category but they wouldn’t object.

Conversely, a more conservative mindset used something like eight metrics. Authority/No Authority Moral/Not Moral, things like that.

So, he posited if you want to sound out someone’s mindset (and you’re willing to live with the repercussions) you can ask: if a man buys a dead chicken from the store, cleans it thoroughly, then fucks it, and then eats it himself…?

I listened in dawning horror, both rapt and disgusted. But into the growing pause I whispered, “No harm…” because it really has no effect on me or anyone else if a man fucks a dead chicken. I don’t like it, I think he’s a weird dude, but like. That’s his dick. But a more conservative person will hear that and object on moral grounds despite not being harmed.

It’s been haunting me all day, so please enjoy.

[wannaliveattheholidayinn]

nils ortega when i catch you. when i catch you nils ortega. nils ortega when i catch you

[bauerntanz]

Die Bielefeld-Verschwörung

Seit 30 Jahren hegen viele Deutsche Zweifel an der Existenz von ... Bielefeld. Achim Held ist schuld. Der Informatiker aus Kiel erfand die Verschwörungstheorie. 1994 hatte er mit dem Spruch: "Bielefeld gibt es doch gar nicht!" die sog Bielefeld-Verschwörung kreiert.

Seit 30 Jahren hegen viele Deutsche ernsthafte Zweifel an der Existenz von … Bielefeld. Achim Held ist schuld. Der Informatiker aus Kiel erfand eine Verschwörungstheorie und ließ die Stadt verschwinden. 1994 hatte der Mann mit dem Spruch: “Bielefeld gibt es doch gar nicht!” die sogenannte Bielefeld-Verschwörung kreiert. Und dann war er vom Erfolg seiner Satire total überrascht. 2019 hatte die…

[th3-pup-3v3r]

Axioms End Ampersand…. Save me Axioms end ampersand….. Save me!!!

[marigold-22]

I actually like this post in concept , I promise I’m not trying to be a hater but most of these proofs don’t work.

1.) does work and also shows the fact that 1+1=2 basically just by definition.

2.) doesn’t work because you defined addition wrong. I’ll give a similar albeit different way of proving 1+1=2 via set theory. First we define the natural numbers exactly as stated in your post, ie 0={} and S(n)=S(n-1) U {S(n-1)} then we say that a set X has cardinality(size) n if we have a bijection from X->S(n-1), a bijection is a mapping of elements from X to elements of S(n-1) such that no two elements of X are mapped to the same element of S(n-1) and every element of S(n-1) is mapped by some element of X. Then we may define addition of n,m by the cardinality of X U Y where X has cardinality n, Y has cardinality m and, this is important and what you missed, X and Y are disjoint meaning they have no elements in common. Then {🍎} has cardinality 1 via the bijection f(🍎)={} and so does {{}} via the identity map. Then since {🍎} doesn’t contain {} and {{}} doesn’t contain 🍎 we have {🍎} U {{}}={🍎,{}} has cardinality 1+1 then via the bijection g(🍎)={} and g({})={{}} we have {🍎,{}} has cardinality 2 thus 1+1=2.

3.) first off N has a monoid structure not a group structure and second off your proof isn’t really a proof, we can equip the set of natural numbers with many, many different monoidal structures in most of which 1+1 doesn’t equal 2, in order to equip the naturals with addition you have to define addition which means that you define 1+1 as 2 which makes the proof self referential. You can’t really give a proof that 1+1=2 via abstract algebra(although if someone can come up with one please tell me I would love to hear it)

4.) not a proof, intuition doesn’t constitute a proof

5.) in the base case of induction you assume 1+1=2 proofs can not be self referential.

TLDR: there are 2 maybe 3 proofs that 1+1=2(that are distinct enough to call different proofs) this post gives 1 of them and attempts 4 more all of which are incorrect.

PS: please keep making these sorts of posts, even if they are wrong sometimes, they’re fun to read

5 Ways to Prove that 1 + 1 = 2

1. The Peano Axioms (aka The Fancy Pants Proof)

According to this system, we start with 0 and define the successor function S(x), which means the "next" number after x. So:

0 is a number.

If x is a number, then S(x) is also a number.

Now, to prove 1 + 1 = 2, we define:

1 = S(0) (the successor of 0).

2 = S(S(0)) (the successor of the successor of 0).

Now, to compute 1 + 1, we use the addition operation defined as:

x + 0 = x

x + S(y) = S(x + y)

Thus:

1 + 1 = S(0) + S(0) = S(S(0)) = 2

So, we’ve just proven 1 + 1 = 2 using the Peano Axioms. 🧐

2. Set Theory (aka The “We’re Just Counting Stuff” Proof)

Let’s take a more "set-theoretic" approach. Set theory is the foundation of most modern mathematics, so here we go:

Consider the set {0}, which contains one element (zero). The successor of a set A is defined as S(A) = A ∪ {A}.

Let’s do it step by step:

0 = { } (the empty set).

1 = S(0) = { { } } (the set containing the empty set).

2 = S(1) = { { }, { { } } } (the set containing the empty set and the set containing the empty set).

Now, 1 + 1 is the union of two sets, so:

1 + 1 = S(0) ∪ S(0) = { { } } ∪ { { } } = { { }, { { } } } = 2

Look at that. Set theory is so neat, it just counts the things and boom, 1 + 1 = 2.

3. Abstract Algebra (aka The “We Can Make This Complicated” Proof)

Let’s prove it algebraically with some abstract structures! Consider the group Z2, the set of integers modulo 2. So, Z2 = {0, 1}, and addition in this group is:

1 + 1 = 0 (mod 2)

Wait, that’s not the right proof. Whoops.

Let’s try again with the addition of natural numbers. We know 1 and 2 exist in the set of natural numbers N. So by the basic group structure of natural numbers under addition, we literally just have:

1 + 1 = 2

In other words, we just used the closure property of N. Let’s move on, before we get lost in algebraic loopholes. 😂

4. Intuition (aka The “Common Sense” Proof)

You can also prove 1 + 1 = 2 the most intuitive way ever. Think about it like this:

If you have one apple 🍎 and you add one more apple 🍎, you’re just holding two apples 🍎🍎. How many apples? Oh, look at that—two. Simple as pie. 🍏

So, 1 apple + 1 apple = 2 apples. Seems legit, right? ✨

5. Proof by Induction (aka The “Math Loves Repetition” Proof)

Alright, get ready for a bit of mathematical recursion. Proof by induction is a powerful technique where we prove a statement is true by showing it works for the base case and then showing it holds for every subsequent case.

So let’s define 1 + 1 = 2 using induction:

Base Case: We know that 1 + 1 = 2 is true for n = 1 because, well... it’s true. 😅

Inductive Step: Now, assume that for some number n, we have n + 1 = n + 1 (this is the step where we take our assumption). We then prove that (n + 1) + 1 = n + 2, which holds by simple arithmetic.

Since both steps hold, by the principle of induction, 1 + 1 = 2 is true for all natural numbers!

So there you have it, folks: 5 ways to prove that 1 + 1 = 2. You’ve got your formal, set-theoretic, algebraic, intuitive, and recursive methods. Math can be rigorous, but it’s also pretty darn fun when you get creative.

[aethon-recs]

Tomarrymort Starter Pack: 10 Recs for Getting Started in Tomarrymort

I've compiled a list of 10 medium to longfic recs that I think represent a great on-ramp to the Tomarrymort ship, as inspired by @sitp-recs’ Drarry for Beginners rec list. These are the fics that I would use to on-board people to the ship — gorgeous writing, superb characterization, and just as enjoyable on the first read as the 20th reread. 

As always, I am stunned by the talent in this ship! I tried to pick a good mix of different themes/tropes/settings, with a focus on elements that make for a good introductory work: the characters are recognizable; the setting skews more recognizable; both characters in the ship are a meaningful part of the story; the ship is central to the story; and the fics are for the most part complete (or updated within the last year). 

(Standard rec list disclaimers apply: please mind all tags and warnings on AO3 before reading; this blog abides by the age-old fandom axiom of don’t like; don’t read; recs are in alphabetical order by title.)

This is Part 1 of a 3-part series — I also have an Intermediate reading list and Advanced reading list coming up for readers who have been with the ship for a longer time.

For now, please enjoy these 1.3 million words of absolutely brilliant Tomarrymort reads that I hope will keep you hooked until the very last word:

*

Tomarrymort Starter Pack Recs

A Dangerous Game by @cybrid (E, 284k, WIP)

Setting: Canon Divergence – Book 5 Premise: If Tom’s diary horcrux gains a body at the end of Harry’s 5th year (instead of his 2nd), and then promptly kidnaps Harry and holds him captive over the summer. Lots of smut ensues. Why I rec it: The characterization is truly stunning — Tom Riddle is undoubtedly a psychopath — manipulative, thrill-seeking, kind of an irredeemable shithead — but he’s also dazzlingly charming when he chooses to be, someone whose presence Harry quickly grows addicted to. Their relationship can get incredibly toxic and fucked-up at times, but Harry has top-notch instincts and can hold his own against Tom. The plot is absolutely gripping, with the threat of (the main) Voldemort, who has set his sights on reclaiming his wayward horcrux, looming in the background. 

A Future Without a Face by @dividawrites (E, 115k, complete)

Setting: Time-Travel (1940s) Premise: If Harry travels back in time to Tom’s 5th year at Hogwarts, and Tom becomes obsessed with the new transfer student and wants nothing more than to possess him every way. Why I rec it: A 1940s time travel fic told entirely from Tom Riddle POV! Divida absolutely nails psychopath Tom — how he quickly gets singlemindedly focused on Harry, how the idea of possessing Harry consumes him, how he has no compunctions about doing completely fucked-up and destructive things to achieve his goals. There is so much tension between them from the start, so there’s not much of a wait to see some hot Harry & Tom action — and the conflict and tension only continues to build and build in dramatic fashion throughout the rest of the fic.

Either must die at the hand of the other by @metalomagnetic (E, 260k, complete)

Setting: Post-Canon Premise: If Voldemort survives the Battle of Hogwarts and is initially kept prisoner in Azkaban, until Harry takes him into Grimmauld Place under house arrest. Why I rec it: This fic is an incredible exploration of Voldemort at his most terrifying. Even if he starts off the fic with his magic temporarily blocked, he is no less powerful without his magic. The force of his personality is powerful enough for him to chip away at Harry’s initial resistance — @metalomagnetic manages to write one of the most charismatic, brilliantly manipulative, and psychologically devastating versions of Voldemort I’ve ever read. Harry ends up in a good place by the end of the fic, but the journey to get there is a roller-coaster of emotions that have permanently imprinted onto my soul.

In Somno Veritas by ladyoflilacs and @lordansketil (M, 158k, complete)

Setting: Canon Divergence – Book 6 Premise: If Harry starts appearing in Voldemort’s nightly dreams during Book 6, and Voldemort becomes obsessed with Harry after realizing he’s his horcrux. Why I rec it: This is one of the most unique fics I’ve ever read in this ship! Every scene is told in alternating POV between Harry’s POV and Voldemort’s POV, so you get to see how every scene unfolds from both of their perspectives. Voldemort is so intense and just as terrible as he is in canon, so his character is not at all sugarcoated, and Harry has so much compassion and heart and manages to fall in love with Voldemort anyway. The writing style is gorgeous, with richly detailed and emotionally-laden prose. Also, one thing that pleasantly surprised me is how funny their banter is! There were definitely a number of times where I laughed out loud in the middle of an otherwise really intense scene. Bonus content: also comes with a lovely sequel that made me melt.

Inevitabilities by @shadow-of-the-eclipse (T, 103k, complete)

Setting: Same-Age AU Premise: If Harry and Tom attend Hogwarts together and go traveling around the world after they graduate. A betrayal leads to their break-up, but after many years, Harry returns to find Tom in Britain, and the two of them are drawn back together again. Why I rec it: An excellent same-age AU with unhinged dark Harry and just-as-unhinged Tom. Their relationship starts out quite dark and twisted and unhealthy — and only devolves from there. The fic ends with the two of them as equals — utterly devoted to each other — but in an incredibly fucked-up way: “He loves Tom like a forest fire; wild and all-consuming, he wants to devour Tom, to claim him, to mark him, break him.” Isn’t that absolutely breathtaking?

love is touching souls (surely you touched mine) by @toast-ranger-to-a-stranger (M, 34k, complete)

Setting: Time-Travel (1940s) Premise: If Harry gets thrown back into the mid-1940s and meets Tom Riddle as a young man just graduated from Hogwarts working at Borgin and Burkes. Why I rec it: When Harry accidentally travels back in time and chances upon Tom Riddle as a fresh graduate, he realizes this is his chance to make a difference. While Harry is only in the past for a brief interlude, he leaves enough of an impression to change the trajectory of Tom’s life. The dynamic between Harry and Tom is rife with tension and witty dialogue, and the story is set during Christmastime, which lends a very festive and heartwarming atmosphere for falling in love with each other.

No Glory by @obsidianpen (E, 254k, WIP)

Setting: Voldemort Wins AU  Premise: If Voldemort figures out Harry is his horcrux when Harry surrenders in the Forbidden Forest, and decides to keep Harry instead of killing him.  Why I rec it: This fic showcases the absolute, terrifying genius side of Voldemort, in a universe where he wins the war and captures Harry at the end of book 7. I am stunned at how skillfully @obsidianpen portrays Voldemort as a brilliant political strategist — the courtroom scene where he manipulates the story and the audience so well stands out as a top 10 fanfic moment in my mind. Harry and Voldemort’s relationship is chilling from the very start, and grows even more unhealthy as Voldemort gets addicted to Harry’s touch due to the presence of the horcrux, but Harry later learns to turn that to his advantage.

The Fire, Burning by @parsimmony (E, 35k, complete)

Setting: Canon Divergence – Book 6 Premise: If Voldemort discovers Harry is his horcrux after Book 6, and kidnaps him to keep him captive by his side in his bed, inside of a lovely greenhouse setting full of friendly snakes on the grounds of Malfoy Manor. Why I rec it: The prose!! I am swooning over the prose! Harry is Voldemort’s captive in this fic, but he is so much more than that — and the emotions that gradually blossom between them have so much richness and depth and are utterly moving that I’m still drowning in the depths of intimacy that were portrayed. Their relationship unfolds in such a gorgeous and unrushed way, and the setting is so unique too — a lush and overgrown greenhouse that’s exploding with exotic plants and friendly snakes around every corner that imbues the fic with a very romantic, dreamy quality.

the pleasure, the privilege by @being-luminous (M, 20k, complete)

Setting: Canon Divergence – Book 6 Premise: If Voldemort is doused with Amortentia keyed to Harry, and starts sending Harry bizarre and gruesome courting gifts, like the bodies of the Dursleys.  Why I rec it: Breathtaking prose! Voldemort somehow ends up more terrifying when he’s trying to woo Harry than when he’s trying to kill him. Every single sentence had me on the edge of my seat, as Voldemort’s ‘gifts’ become more elaborate and devastatingly dramatic, until Harry basically has no choice but to respond to his overtures. The ending is incredibly clever in how it parallels certain plot elements of book 6, with an added Harrymort twist. 

The Untouchable by @treacleteacups (M, 75k, complete)

Setting: Canon Rewrite (Books 1-7) Premise: If Harry starts out his first year a little bit more suspicious and a little less wide-eyed and guileless, and subsequently gets sorted into Slytherin. He has many of the same encounters with Voldemort along the way as he does in canon, but his interactions with Voldemort will end up leading him down quite a different path. Why I rec it: A snappy, fast-paced full canon rewrite that still manages to fit in all the essential Tomarrymort plot points, between Horcruxes and Hallows and the major events of books 1-7, in a compact 75k words that doesn’t at all feel rushed. It’s a delightful journey following Harry’s character evolution from an overlooked, peculiar child who relies on wishy-washy wish magic to a confident (and still endearingly peculiar) young man who can challenge and hold his own against the great Lord Voldemort. Voldemort’s obsession with Harry deepens with every encounter that they have, as he finds ways to continually insinuate himself in Harry’s life and his mind and his dreams.

*

[catgirl-lucy]

Ultrafilters & Ultrapowers

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 non-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!

That's all I had to say.

Bye!~ Have a nice day.

[genquerdeer]

The only distinction that would make sense is that Vania is for game with focus on RPG elements (i.e. numerical statistics that you increment by playing and not just by exploration) and Metroid is for gamnes that don't.

Because let's face it, "Metroidvania" is not actually named after Metroid and Castlevania - it's named specifically after Super Metroid and Castlevania: Symphony of the Night. And SotN being an action RPG is easily it's largest difference from SM.

do you prefer the metroid or vania side of metroidvania

I don't think it's useful to taxonomise them that way. I've never seen anyone produce a definition of what the distinction between "metroid" and "vania" is that didn't end up with at least one actual Metroid or Castlevania game on the "wrong" side of the dividing line, much less one that could fruitfully be applied to the genre as a whole.