#model theory

[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 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.

[algebraic-dualist]

One of the frustrating aspects of research literature is that filling in the details to understand some result you wish to extend can be an absolutely gruelling process. I've been labouring now for a week and a half to merely understand why two technical conditions map to each other under a functor. I think I am making progress, in that I feel that vague shifting, perhaps shuddering, sensation in my brain where the frustration wall starts to become porous that usually precedes full understanding.

[pv1isalsoimportant]

Professors will skip a nontrovial part of a proof and then write this

like thank's for reminding me of what \aleph_0^+ is i guess.

[m---a---x]

Inspired by all the newly created communities i have also created one about the topic closest to my heart: Foundational Mathematics

It is inteded for all types of posts about and from people of all kinds of backgrounds interested in the topic.

Please share with anyone you think might be interested. If you want to be added comment on this post, so I can add you.

[sqrt-48]

Language

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.

[vacuous-truth]

Model Theory has become my new interest just because of how often some very self-explanatory statement is actually wrong, and how often instead the useful statements look like something the mad hatter would justify.

Like in the “sentences that shouldn’t be allowed” category. If a theory has a infinite model, then it has a countably infinite model which leads the skolems paradox that a countable model of set theory still contains “uncountable” sets but this can still all neatly be explained without contradictions.

Or the Rado graph which satisfies the same first order sentences as “most” finite graphs (with most in this case meaning the probability goes to 1 as the size of the graphs increase to infinity).

Yet first order logic isn’t strong enough to prove that an infinite linear order doesn’t hide the rational numbers somewhere inside it.

Then also the way you can talk about logic through the lens of set theory or algebra or topology or game theory and all these approaches can come up naturally when asking other questions. Love mathematicians creating a bazillion types of structures and then also showing how they’re all specialised versions of eachother

[crystaltoa]

And now for something completely different.

This is the ADHD Teapot. I made it in a ceramics class a few years ago. I use it to explain executive dysfunction to people who haven’t come across the term before (and those who think of ADHD mostly as Hyperactive Eight Year Old Boy Syndrome).

So, most people’s brains are like a regular shaped teapot with a single spout. Let’s say that your time, energy, focus etc is the liquid you have in the teapot. Your executive function is the spout, that directs the tea into the specific cup you want to fill-aka the task that you’re meant to be doing. Spills happen occasionally, but generally most of the tea goes in the right cup.

If you have executive dysfunction, (a symptom of ADHD, trauma, autism, schizophrenia etc.) you have multiple spouts going in different directions. You can try pointing one of them at your chosen cup and you will probably get some liquid in there, perhaps you will even fill it right up (finish the task). But meanwhile, tea is also pouring out of several other places and not going where you want it. If you have another container nearby, perhaps some of it will end up in there. But quite a lot of it is going to end up on the floor and accomplish nothing.

And at the end of the day you’ll have filled one or two cups ( or sometimes not even one) compared to the five or six that somebody with the same sized teapot (but only one spout) has filled, and everyone wonders why you’re so bad at getting tea poured, and why you make such a mess in the process.

One day I’d like to spend more time learning pottery and create a really technically good fucked up little adhd teapot. But that’s a long way off since i currently live in the outback and the nearest pottery workshop is some 400km away. But I figure that for now, it might be a useful or interesting metaphor to somebody even in its rough draft form.

This post is the cup I filled instead of cleaning my house btw.

[geopolicraticus]

Finding Nomothetic Rigor in History

Relative Uniqueness.—Schopenhauer’s claim of Herodotus’ historical categoricity demonstrates that there is more than one way to find nomothetic rigor in history. In the absence of the form of abstraction employed in natural science, history unwittingly makes use for the forms of abstraction employed in the formal sciences. The natural science approach would subordinate the unique events of history to general concepts, which would transform history into a nomothetic discipline if only these methods could be applied in this context—and if history would be a science, then this is what it must become. Schopenhauer, however, denies this of history, as Dilthey, Windelband, Rickert, and many others also have denied this of history. The approach of the formal sciences is to conceptualize unique events as tokens of a type that are not subordinated to a general concept, but are rather exemplifications of a formal concept. If Schopenhauer was wrong, and history is not categorical, then there is no unique model of history and we may need more than Herodotus to understand the totality of history. When an unprecedented event disrupts the historical process and redirects events onto a new course, Herodotus may no longer serve, but other works, works that describe a world in transition, may provide the models needed. It may yet be the case that each era of history, and each kind of era in history, has it own unique model—which in this case means one paradigmatic history for each period and for each geographical region. This is more fragmented than Schopenhauer’s historical categoricity, but also more adequate, and we can still obtain a model for the totality of history from the conjunction of the several models.    

[flameshadowconjuring]

'Our resources are depleting. I'll quickly duplicate them with Banach-Tarski.'

'It won't work. The geometers have locked us in Solovay's model. Everything is measurable, you can't use the axiom of choice here.'

'Hah! We may be barred from an exterior theory, but we still have one out. Are you all definable?'

'Yes...'

'Then we all have representations in the inner model, which is well-orderable. The axiom of choice holds there globally. I will transform us into ordinal-definable representations, transporting us to the inner model. Then, we use Banach-Tarski. They will rue the day they crossed us algebraists!'

[trfur]

Here's another weird model of ZFC relating to the axiom of regularity that fucked me up when I learned about it. The axiom of regularity implies that there can be no set {x_n | n ∈ ω} such that x_{n+1} ∈ x_n for each n ∈ ω.

Let's construct our model. Add to the language of ZFC countably infinitely many constants c_0, c_1, c_2, ... and let Γ be the set of sentences Γ = {c_{n+1} ∈ c_n | n ∈ ω}. We will use the compactness theorem to show that there is a model of ZFC ∪ Γ.

Let Δ be a finite subset of Γ and Let J be any model of ZFC. Since Δ is finite, there is a maximum k such that the sentence c_k ∈ c_{k-1} is in Γ. Add to J the definitions, for each n ≤ k, c_nJ = k - n, and for each n > k, set c_nJ = 0. Then for all 1 < n ≤ k, c_n = k-n ∈ k-n+1 = c_{n-1}, and so J is a model of ZFC ∪ Δ.

Thus, by the compactness theorem, there exists a model of ZFC ∪ Γ.

This is very surprising, and at first glance seems to contradict the axiom of regularity! But what it really means is that the sets x_n from the first paragraph can exist, but they cannot be gathered together in a set.

downward lowenheim-skolem is so fucked up to me. what do you *mean* there's a countable model of first-order set theory

[yourstrulynobody]

Frost and Nova from "THE NEW SUN AND MOON!"

{ !! NOVAFROST !! }

Here is the ref I used for it :D! I seriously love their designs—its pretty!!!!

This is based off FNAFSun's and FNAFMoon's original posters in Security Breach :D!

[pencildragons]

[rolls over to face you at the sleepover] bro do you think any of the lyctors ever used AD (alecto's dead) as a way to reference the passage of time and nearly gave john an honest to god heart attack because he thought they'd suddenly somehow remembered the christian calendar system. no hey wait stop pretending to be asleep this is important dude

[pv1isalsoimportant]

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.

[angryducktimemachine]

Aromanticism Zine but it's just my incoherent thoughts.

[romancerepulsed]

aspec terms for beginners!

since it's trending right now, i feel like it might be helpful to clear up some basic aspec (but particularly aromantic, as we are the center of attention currently) terms. if you have absolutely any questions, i would be happy to answer, either in the replies, dms, or my inbox!

★・・・・・・★

the split attraction model (SAM): a model of human behavior that posits that, for some people, romantic and sexual attraction are not the same.

[most often this will come in the form of someone being aspec on one axis and allo (not aspec) on another. for example, a biromantic asexual may be romantically attracted to two or more genders, but sexually attracted to none. some people may even use SAM for allo identities– a bisexual lesbian may be sexually attracted to multiple genders, but only romantically attracted to women (note that this is not the only way that someone can be an mspec lesbian, just one way!). the SAM does not apply to everybody, not even all aspecs! there are non-SAM aros, for instance, who do not differentiate their aromanticism from their sexuality.]

aspec: a collection of queer spectrums centered around the lack of a certain attraction or identity. the most common spectrums under the aspec umbrella are asexual, aromantic, agender, and aplatonic, though there are many other ways to be aspec.

asexual: experiencing little to no sexual attraction.

[aces can still have sex– whether its because they experience some amount of sexual attraction or they just want to participate in sex because they find the act appealing in some other way. that being said, there are still plenty of aces who have not and will never have sex. it is a spectrum.]

aromantic: experiencing little to no romantic attraction.

[aros can still have romantic partners– whether its because they experience some amount of romantic attraction or they just find relationships appealing in some other way. that being said, there are still plenty of aros who have not and will never be in a romantic relationship. it is a spectrum.]

agender: having no gender or little relation to any gender.

aplatonic: experiencing little to no platonic attraction.

[similarly to aros and aces, apls can still form friendships if they so desire– whether its because they experience some amount of platonic attraction or they find friendships appealing in some other way.]

aroallo: combination of aromantic and allosexual– allosexual being someone who fully experiences sexual attraction. an aroallo, then, is someone who is aromantic but not asexual. aroallos often do not have a standard relationship with sex due to its romantic connotations and the stigma against loveless sex. someone having sex with someone else they do not love does not inherently make them aroallo, much in the same way that having a nonsexual relationship with a partner doesn't inherently make either participant asexual.

aroace: someone who is both aromantic and asexual. because aro and ace are both spectrums, an aroace may still experience some amount of attraction on either or both of those spectrums, or they may experience attraction of some other kind (platonic, tertiary, etc.), and that attraction may be only for a certain gender or genders– these are known as oriented aroaces.

queerplatonic relationship: a type of relationship that is defined only by the people within it. i have a post dedicated to explaining this in larger detail.

partnering: an aspec (usually aromantic) person who has and/or desires to have a partnership or multiple partnerships– romantic, queerplatonic, or otherwise.

non-partnering: an aspec (usually aromantic) person who has no desire to form a partnership of any kind.

romance/sex/plato favorable: an aspec who desires or would not reject a romantic, sexual, or platonic relationship. they are also generally not particularly bothered by seeing these relationships in their day-to-day.

romance/sex/plato repulsed: an aspec who does not desire a romantic, sexual, or platonic relationship and generally does not like seeing those relationships in their day-to-day. [x] repulsed people are not necessarily judgemental towards people who desire or participate in those relationships, they just do not desire them for themselves. repulsion often takes the form of discomfort or annoyance. [x] repulsed people are not necessarily cruel sticks-in-the-mud– they are perfectly capable of being respectful, and they very often are. repulsion does not always stem from trauma, though it certainly can.

romance/sex/plato positive: not to be confused with favorability, [x] positivity is the belief that romance, sex, and platonic relationships are human rights that should be supported and uplifted. someone can be [x] repulsed and [x] positive at the same time, because favorability/repulsion revolves around the self, and positivity/negativity extends to others.

sex/romance/plato negative: not to be confused with repulsion, [x] negativity is an inherently judgemental and harmful ideology. most commonly in the form of sex negativity, these ideologies are centered around the opposition to or personal judgement of people who engage in romance, sex, or platonic relationships. sex negativity in particular is embedded in western white supremacist societies and it is important for aspecs not to play into that.

those are the basics, but i have more information below the cut!

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> how are aspecs queer?

aspecs are queer because "queer" does not only mean LGBT. queer theory is about far more than just LGBT people– though they are undeniably a large part of it– queerness is any subversion of the traditional cisheteronormative standard. this includes things that cishets may take part in/identify with, because you do not have to be LGBT to subvert those standards. cishets who are gender-nonconforming are queer, for example. a good rule of thumb is that if you have to explain what you whole deal is to cishets, you're queer. queer does mean strange, after all.

traditional cisheteronormative conceptions of attraction, gender, and relationships do not account for aspecs. it is expected that everyone will one day form a traditional partnership with one other person, and that relationship will include sex (even if only for procreation, under some dogmas). virginity past a certain age is seen as a point of shame and something indicative of a larger problem in someone– in men, a red flag even. people past 30 without a relationship are pitied. our economic structure is build for couples and families– it's near impossible for someone to live comfortably alone. romance, friendship, and love are placed on a pedestal, treated as the meaning of life, the best thing anyone could ever experience. "love is the point of everything," as many posts on this site like to claim. people who reject these ideas are undeniably queer.

> i can get behind aros and aces, but the whole "aplatonic" thing feels like a stretch to me. how is not having friends queer? "platonic attraction" isn't even real.

aplatonicism is more than just "not having friends," and many apls have friends anyway, much in the same way that aros can date and aces can have sex. someone who does not have friends is not inherently aplatonic, they only are if they identify that little-to-no platonic attraction in themselves and choose to label themselves that way (just like how virgins aren't inherently asexual). still, apls who don't have friends exist, and they are all queer. what is a greater subversion of traditional cisheteronormative relationship structures than an outright rejection of what's seen as the most basic, fundamental relationship our culture has to offer?

you may not feel that platonic attraction is a distinct phenomenon in your own experience, and that's fine! ultimately, a lot of aspec terms exist for the utility and comfort of aspecs themselves. the SAM isn't for everyone, and platonic attraction isn't for everyone either. you do not have the authority to tell people what their own experiences are, nor should you care.

> i think it's sad that you're limiting yourself with these labels. you'll find someone one day!

for the broad majority of aspecs, our identities are not self-disciplinary, nor are they necessarily permanent. all queer people are capable of misunderstanding their identity or having a fluid identity– it is not a problem unique to being aspec. that being said, a lot of us may always be aspec and completely happy with it. being aspec is not a tragedy. the only thing i don't like about being aromantic is the judgement i receive from other people about it. non-partnering aspecs are not "missing out" on anything, because we don't even want the things we're rejecting in the first place. many of us are romance/sex/plato repulsed and are far more happy engaging with the world and with other people in different ways, because there is so, so much more to life than relationships, and it's wrong to presume that relationships are universally fit for everybody. telling an aspec that they'll find "the right person" one day is no different from telling a lesbian she'll find "the right man" one day. there is no "right person" for an aspec just as there's no "right man" for a lesbian. a lesbian is not "missing out" on a heterosexual relationship just because it's culturally perceived as superior and more fulfilling.

[disclaimer before anyone tries to do a "gotcha," i'm talking about a lesbian who is fully not attracted to men in any way. it's not like homophobes know the intricacies of gender identity and nonconformity as it pertains to homosexuality anyways.]

lastly, i wanna give a special shout out to the loveless aros and the relationship anarchists.

loveless aros are those who either feel little-to-no love as they understand it, or they are someone who supports the de-centering of love. they're worthy of a whole post of their own, but in summary: the loveless experience is all about finding joy in yourself and the countless things our world has to offer that are not dependent on the vague idea of love.

relationship anarchy is another concept worthy of its own post, but in essence it's an ideology aimed at abolishing the standard hierarchy of relationships (in the USA, depending on who you ask, its typically friendship < family < romantic partnership or friendship < romantic partnership < family) and allowing everyone the autonomy to define their relationships for themselves.

if i made any mistakes, let me know! and of course i'm willing to answer any questions anyone may have. :-3 thanks for reading my long ass post!