#first order logic

[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

[starry-bi-sky]

i have been unmedicated for the entirety of spring break and thus have had little interest in writing this down, but i have been thinking about this for the entire week (as well as a dpdc clone danny au that resulted in it becoming its entirely separate batman au that includes a teenage vigilante bruce wayne, an ocarina, and me entirely incapable of making a batman au without making bruce dirt poor but we're not talking about that) and so i've finally went 'fuck it' and forcibly grabbed my laptop. I will get this done in one sitting even if it kills me.

BUT. This is about neither clone^2 danny nor about who i am calling Ocarina Batman. This is about my Danyal Al Ghul Au and more SPECIFICALLY it's me thinking about his relationship with Sam and Tucker specifically.

Tucker and Sam? Adore this asshole (affectionate) with every fiber of their being. And it is very much a reciprocated feeling, but Danny's thoughts will not be delved into much other than he would kill for them.

Tucker? The only person currently capable of getting a deep, loud, belly laugh out of Danny. Sam can get him to smile and to laugh, but it's the kind that's a chuckle-under-the-breath. The quiet, looks-down-while-huffing laughter. Snorts once with laughter and then grins stupidly.

But Tucker? Tucker can crack a slew of stupid jokes and Danny will be incapacitated for the next five minutes because he's laughing so hard that he can't breath. He lands one well-timed pun or quip and Danny will be close to tears. His laughter is their favorite sound in the whole world.

Sam is lowkey jealous of this ability, and she's gotten a belly laugh out of Danny a few times. But alas, it is Tucker who wields this power and has gotten it the most times out of the two of them.

-

They're also both physically affectionate with Danny as much as possible. It started roughly around when they were 12-ish, a year since they befriended Danny, and they noticed that he sought after touch but never seemed to initiate (and was in some ways repulsed by it). They started slowly being more touchy with him. Hooking a finger around his to lead him somewhere, tapping his wrist, looping arms. Little touches, grabs, etc, to get him used to it, and once he started doing it back they started increasing it.

It's gotten to a point where he will now just. Lay on them. Like a lizard sunbathing on a rock. Leaning on their backs when they're sitting in class before the bell rings, his chin on their heads. He'll talk about anything with his arms looped around their shoulders.

If they're sitting on a couch at either of their houses, he'll lay his legs on theirs. Him and Tucker will press their feet against the other's and try and push against them (newsflash: Danny always wins, Tucker claims its the ghost strength but Danny's been winning since before his accident)

-

Naturally, both Sam and Tucker know where Danny keeps his weapons on his person, and are allowed to grab them off of him if they need it. His only requirement is that they don't lose his weapons if they take it and forget to return it immediately.

They both understand how big of a thing this is from Danny, and so they do their best to treat his weapons with a lot of respect and care because they know its his way of saying he trusts them.

-

Sam and Tucker are so fond of Danny it's insane. Like fr. That's their goddamn best friend, and they are so protective of him. Emotionally, physically, you name it. They will tear the head off a grown man if they need to, Danny's had scars since he arrived in Amity Park and Sam and Tucker both are going to find the person who put them there and make them pay for it.

One time, Tucker overheard a bunch of upperclass girls speaking nastily about Danny and about the rumors surrounding him, calling him names like 'freak', 'monster', etc. Danny was with him and heard it, and seemingly appeared unbothered by it, even telling Tucker that he was used to such rumors.

Tucker was so furious that hacked into the school system later that night and tanked those girls grades. They were kicked out of their clubs and had to go to mandatory tutoring for the rest of the year. He made sure to leave some way of letting them know it was him who did it.

And Sam doesn't like using her money for things, doesn't like abusing that wealth. So instead, whenever her parents talk bad about Danny, she causes a media incident that has her parents scrambling to deal with. She does something wild, outrageous by her parents' standards.

She heard some boys on the basketball team making fun of Danny once, similar to those girls had. She kicks up a fuss about something eco-unfriendly at school and forcibly holds a protest on the same day of the big home basketball game, forcing them to cancel the event and reschedule to a visiting school.

She anonymously donates money so that there's new uniforms for the team but oops! Looks like she "forgot" to donate enough money for them to get uniforms for all the team members, and strangely enough those boys in particular didn't get them! Looks like they'll have to wait until more money gets donated for the basketball team to get their new, nice uniforms. The old ones look so ratty in comparison, right?

And since the football team gets most of the sport money, that might just take awhile. And if (and when) they kick up a fuss? oops! Off the basketball team you go, :) such unsportsman-like behavior is unfit for the team.

(The only good thing about how corrupt the school system is is that she can use it to her advantage too.)

The both of them know that Danny suspects them for the sudden misfortune falling on these people, but he doesn't call them out on it. He's kinder than he used to be, but not kind enough to vouch for people who speak badly of him. Sometimes, he might just congratulate them on not getting caught.

Because Danny is their wonderful, hurt friend with a "slightly" Blue and Orange Moral code, and enough scars that people have been calling him a criminal (and worse) since he arrived in Amity Park when he was ten. And they'll be damned if he gets hurt anymore.

[buttercupshands]

had a sudden inspiration wave hit and also decided to make this a practice and a study of stuff

[sysig]

Justify away (Patreon)

[jucomx]

I answered yes but there are good justifications for all three answers, where strictly yes is possibly the weakest one:

NO:

Assuming a prerequisite of working within arithmetic with common expansions by definition of the language, but no additional expansions by definition, you are writing a nonsense expression. Those symbols do not have meaning written in that way within the standard framework, so the answer from the logical language of arithmetic is no this is not ok.

ORDER OF OPERATIONS:

However, by defining an order of operations, we give that sequence of symbols meaning. An order of operations is an expansion by definition of the logical theory arithmetic (or any logical theory in a language with function symbols for that matter), we provide a way to uniquely map any syntaxless sequence of symbols to a term in the language of arithmetic.

In particular an n-term induces a formula (in a language expanded with f as n-ary function symbol, this can be a family of formulae but iirc a single formula phi(x*,y) which is a functional class in the unexpanded language will always suffice) phi:"forall x* (f(x*)=t(x*))", where x* is an n-tuple of variable symbols. This formula uniquely determines an assignment for the n-ary function symbol f in any model of the theory of arithmetic expanded by this formula .

By choosing an order of operations to use in arithmetic, the american elementary school standard order for example or even literally just left to right order, and then any expression that can be interpreted by the order of operations is a valid expression to write. However this kind of goes against the spirit of the question because then 2^1^2=(2^1)^2 is also interpretable but not ofcourse equal to 2^(1^2).

YES:

The most fun way i can think of to answer your question though is to take ambiguous formulae and make them an expansion on the notion of "terms" in first order logic. Concretely this is done by making them relation symbols in an expansion by definition of any given theory. However we define these symbols in such a way that they may be used notationally as terms. In fact we could build first order logic from this conceptual expansion but it is orders of magnitude easier to just introduce this concept through existing concepts in first order logic.

First we define what an ambiguous expression is. An ambiguous expression is an equivalence class of terms determined by expansion to an equivalence relation of the following recursively defined relation, with f,g function symbols, x*,y* tuples of terms, and t,s terms:

f(x*, t, y*)~f(x*, s, y*) <= t~s

g(x*,f(t, y*))~f(g(x*,t),y*)

Notice that this relation corresponds to the informal relation "are the same when removing brackets", but more general since it applies to function symbols that don't use inbetween notation too. We can extend the notion of ambiguous expressions by first extending our language to duplicate all function symbols, denoting f' as the duplicate of f. Adding the formulae "forall x* (f(x*)=f'(x*))" to the theory makes it an expansion by definition.

We then expand our definition of ambiguous expressions to these f' by only including the first equivalence for these function symbols. Informally f is a function "without parentheses" while f' is that function "with parentheses".

An essential property of these equivalence classes and a good exercise for the reader is that every equivalence class is finite, and the variable symbols appearing in any two equivalent terms are the same. (hint: the order of terms from left to right is the sane on the LHS and the RHS of the elementary equivalences.)

Now that we have these equivalence classes on terms in our expanded language, fix any n-ary relation symbol (including "=") and any n ambiguous expressions. Introduce an m-ary relation symbol , where m is te sum of the amounts of variable symbols in representants from each of the n equivalence classes. In this way, add a new relation symbol for any relation in the initial language and any corresponding tuple of ambiguous expressions.

Fix any newly added relation symbol R([t]*), where R is an n-ary relation and [t]* consists of n ambiguous expressions, say R([t]*) is m-ary. With v** an m-tuple partitioning as n different tuples v*_i each corresponding in length to the amount of variables in elements of [t]_i, we can write a property axiomatizable as some phi(v**). is the elementary formula of the property 'for all tuples r*,s* of terms such that r* and s* satisfy componentwise inclusion in [t]* we have R(r_i[{v_i}*]) ^ R(s_i[{v_i}*])'.

Notice the finiteness of all ambiguous expressions as equivalence classes guarantees axiomatisability. We can just take a conjunction of the elementary formula part of the property over the finite amount of pairs of tuples of terms componentwise included in [t]*, which is of course still an elementary formula (although it can get rather large).

Finally we can make an expansion by definition in this expanded language. For every newly added relation symbol R([t]*) and corresponding m-tuple v** of distinct variable symbols, we add the formula "forall v**(R([t]*)[v**]<-->phi(v**)", with phi the elementary formula described above. We can replace the logical and with a logical iff in the axiomatisation phi to get relation R*([t]*), which checks "well dedinedness" rather than "well defined truth".

Notice that everything with regular terms injects neatly into ambiguous expressions through the duplicated function symbols having singleton equivalence classes. Well defined truth of an expression in that case just means truth since the well definedness is trivially satisfied.

To conclude this text we look back at one of the examples you gave, the ambiguous expression 3^2^2 is the equivalence class of terms {(3^2)^2, 3^(2^2)}, notice this is ^(^(3,2),2)~^(3,^(2,2)), and the ambiguous relation 3^2^2=81 evaluates to true in natural arithmetic (therefore 3^2^2=*81 certainly does).

This is something i just tought up so as far as i know it is not common consensus at all, it has basically no practical applications as far as i'm concerned. But it does provide a framework for your expression to be sensical and evaluate to truth. I think this is probably some subcategory of an application of some existing subject but oh well...

[petr1kov]

one thing i'm noticing about most cobra kai discussions that i'm seeing online is that many people seem to not have watched the karate kid movies before watching the show -- which, on the one hand, is understandable given that the show does provide enough context to stand on its own (and that many fans are young enough to have missed the original movies), but on the other hand, there will always be some things that you can only really get if you watched the movies.

i, however, dutifully decided to do my karate lore homework before starting the show, so i'm aware of the difference this should make in the experience. like, for instance, the fact that daniel opposed the creation of cobra kai less because of his personal rivalry with johnny and more because an insane coked up millionaire terrorized him and mr miyagi for weeks when he was a teenager and tried to groom him into having a villain arc on karate kid 3, specifically through recreating cobra kai but doing it in 'the right way' and tricking daniel into joining it, all with the goal of corrupting daniel and making him suffer a humiliating defeat in order to bring the dojo back and spread it throughout the country like some saturday morning cartoon villain. an experience which shook daniel up more than having a duel to the death with a karate assassin who tried to kill him and his girlfriend with a knife. so when he sees his old school rival trying to reopen the same dojo, he's not into it lol

[one-pump-chump]

can you imagine what an absolute dumpster fire gta v would be if simeon had sent lamar to michael’s house instead of franklin?

it’s fairly obvious that franklin is written to be michael’s perfect mouldable protege in the beginning: tough upbringing, seemingly doomed to a life of crime, with a level-headed pragmatism and empty, materialistic streak. and what is lamar but the trevor to franklin’s michael?

so how would michael react to someone who reminds him of his psychotic old running buddy so fiercely? fuck, how would trevor eventually react to this young, dumb and full of. uh. nerve? and cockiness, and unpredictable violence?

i always found it compelling how trevor treats franklin a little bit like Michael’s Newest Victim when they first meet. how much more insane would it be, if he saw himself in him?

[craptainkirk]

Part of the reason i like captain kirk as a character is that he will put everything on the line on the CHANCE that something can be done. He's always thinking that there has to be a way, and if not they'll create one. He's also so feelings-oriented and will fall back on his hunches when needed and his hunches are in fact treated as something sacred. I'm the same way in every regard and it's something i've always been criticized heavily for (I've constantly been told that i don't think things through and i'm chastised for being reckless). But that's how captain kirk is and he's treated as if that's his own unique genius. He's respected for it and we can see how if he weren't to take the risks that he does then they'd all be doomed in so many situations. This isnt really well written but im just thinking out loud here... ok... End post

[to-be-a-dreamer]

In case anyone else was wondering, I did the math and, to reach a speed of 5000 mph by going up in a plane and jumping out or something, you would have to free fall from a height of approximately 160 miles, which is in the thermosphere (not breathable air) and doesn't leave you with enough distance to decelerate safely so it's not really a viable method. Yes I took a break from the action movie show to do physics don't worry about it

Scratch work for reference

[jasper-unofficial]

not gmmtv updating the 2025 lineup playlist to place the trailers in order from most to least viewed

[taffywabbit]

Thank you for spreading that thing about voting. The rhetoric I've been seeing from a lot of the left, at least on here, has mirrored 2016 really closely and it's getting me extremely nervous.

yeah considering how many self-destructive doomer-y "it's not even worth voting, nothing but DIRECT ACTION can save us now" posts i've seen cross my dash over the past couple years i figured it was probably smart for me to spread it when i saw it going around again. i'm not american myself, but i have a lot of loved ones down there who really especially cannot afford a republican win this year, and i'll do whatever little i can to help them stay safe-ish. (not to mention i DO still share a land border with the US, and a vicious autocrat god-emperor would probably not be a great next-door neighbor for me either)

[july-19th-club]

favorite type of villanous characters are the ones whose motivations boil down to

like they dont even have to have any stakes in the broader situation. they just have take pleasure in destruction

[rippledawn]

Had a fun time this afternoon

Was hanging out with some of my friends who are also mathematicians, and as mathematicians do we of course started spontaneously talking about deep philosophical topics like what is information, and does mathematics exist independently from intelligent life to interpret it

Could not find a nerdier conversation topic, but we had a fucking blast

God I love being a mathematician

[sonknuxadow]

sorry for randomly talking about monster high characters as if my sonic followers know who these people are. Anyway. im not actively opposed to toralei and clawdeen dating in the comics its really cool to see clawdeen get a girlfriend in official media after her being a lesbian was a popular headcanon for so long . and also toralei and clawdeen are 2 of my favorite characters so getting content focused on both of them at once is a win. but i AM kind of disappointed/confused that the new comics seemingly just have them dating from the start without showing how they got there? if it was a random side character that clawdeen was already shown to be friends with i wouldnt really question it but everyone kinda couldnt stand toralei back in the original g1 media so . what happened here

[skrunksthatwunk]

just finished the original devilman manga. wuh!!

[aroaessidhe]

Hi!

I just read archivist wasp and loved it! but apparently there are a bunch of other books in the same universe and I was wondering what reading order you'd recommend :)

Also I love your blog and art!

-E

very comedic timing, I literally just started a reread yesterday lmao. Also, thank you!

OK so the main things are three books + a novella + a vaguely connected middle grade, and some extra stuff online (which I'll get to after these). Here’s publishing order for the books, which is a good reading order:

Archivist Wasp

Latchkey - book 2 in the (unfinished) wasp trilogy, following her & the Upstarts & our ghosts (and more about their backstory)

Firebreak is a standalone narratively, but has some significant connections. It’s set in a dystopian city before the apocalypse that happened long before Wasp’s time.

Jillian Vs Parasite Planet is a middle grade space adventure set before Firebreak (the closest to recognisably our present), and is the least connected to everything else (tonally and literally)- but it has a character/piece of technology that is significant in:

Flight & Anchor, a novella prequel to Firebreak (you would understand this if you read it before JvPP & maybe F, but would miss small details/references)

I would consider Archivist Wasp > Latchkey > Firebreak, or Archivist Wasp > Firebreak > Latchkey, then everything else more or less in whichever order you want?

I haven't tried reading Firebreak inbetween (until right now - I was just gonna read AW&L but decided to slip it in there too after I saw this ask lol, just started Firebreak) so I can't vouch for whether it'd be more rewarding to read F or L first, but I like the idea of that. If Firebreak is more accessible to you (likely, as it's better distributed) then I think go for it.

---

The short fiction which is all online, mostly came out just before or after Firebreak, and will make sense if you’ve read at least Archivist Wasp & Firebreak!

Pathfinding - short story about our supersoldiers (free)

Last Chance - set in a different town a little before Archivist Wasp, following a side character in AW/L (free)

Archivist Wasp Myths: Ember Girl at the Crossroads , The Story of the Harvesting-Knife and How It Came to Us Out of the Before-Time Into Now, Catchkeep's Favourite Children, Carrion Boy and the Crow, The Story of Why Catchkeep and the Chooser Hate Each Other In the First Place, Ember Girl & Carrion Boy Fight the Metal Men - essentially the full versions of some of the mythological stories in Archivist Wasp. You’d get the most out of these reading them after the main 3 books. (all on patreon)

FAQ - this is like. a separate short story that you could argue is set in the Firebreak era lol (patreon)

Catchkeep - this is the draft version of the third book in the Archivist Wasp trilogy, currently being posted on patreon, though it’s only a few chapters in and it’s been radio silence for a while. (When (🤞) this is finished I guess I’d say you could choose to read the trilogy first then everything else)

and also:

Archivist Wasp deleted scenes and commentary (patreon)

Latchkey deleted scenes and commentary (patreon)

unpublished short story that inspired AW (patreon)

This post outlining connections between the first 4 books and some of the short stories (free)

the patreon is just $1 for access to everything! There's a couple other unrelated stories there too. Hope this makes sense!!!