Corner of Constructs and Calculations

Mathematics, worldbuilding, and whatnot

[anremithrl]

Hello. Got rid of my introduction post because someone did remind me of common sense and being safe on the internet and not volunteering personal information. I am slightly sad because I did like my intro.

I think I'll go back to being a mysterious lizard on the internet, I suppose.

Hello. I am me. I mostly post about Mathematics and the Cosmere, but as this is my main blog, also any other random shiny posts I see and wish to reblog.

I have 13 sideblogs! My three favourites at the moment are @the-ordinals, @the-first-gem, and @nightblood-type-iv-entity, so check them out!

[anremithrl]

Some math things I have said on my blog, (AKA, posts I am proud of).

No constructive LEM

Adding Perpendicular Numbers

What is a Ring? (and what is a group?)

Glossary of Math Notation

Also check out my sideblog @the-ordinals for more like these.

[spookthelilguy]

Gonna charge my personal Chromebook back up and then factory reset it so that I can access my burner Gmail more easily

[anremithrl]

Switch operating systems while you're at it. Consider the freedom.

[spookthelilguy]

teach me how in around an hour and we got a deal

[anremithrl]

You do understand you just asked me, a human of the sort that really enjoys teaching people about things they are interested in (admittedly a common variety here), about something I am interested in, right?

(Very gladly, yes. You have triggered the proverbial unskippable cutscene)

What kind of Chromebook do you have?

[peachesmendes-deactivated201903]

“don’t double text”

i will deadass send you ten messages in a row on completely different topics

[thememedaddy]

[spookthelilguy]

Gonna charge my personal Chromebook back up and then factory reset it so that I can access my burner Gmail more easily

[anremithrl]

Switch operating systems while you're at it. Consider the freedom.

[incorrectfantasyquotes]

Adolin: why is Shallan toxic?

Kaladin: well she attempted murder and jumped someone in the club

Adolin: okay and how is that toxic?

[alethkars-true-king]

I... don't know. I don't know that people can change that much. I guess I changed... but... *shivers, breathing quickens* but I just... I can't...

[story-cryptic]

*concerned buzzing* It's alright, little king... you do not have to be around him. He is not seeking you out, is he? You can... just stay away and let it be, until you feel ready. There is time.

[alethkars-true-king]

I still know he's there though! I can feel him... everywhere. It's like he's watching me all the time. And everytime I fall asleep, I... he's there. Trying to kill me. Trying to kill Father. I wish he hadn't come back. I wish he'd stayed dead.

[story-cryptic]

Gav... I... mmmm... you are like your father. Scared of things that are real for you.

It will take time to understand truths, and sometimes I think human bodies do not cooperate with the mind. There are... many people in Urithiru whose bodies tell them they are in danger when they are not. And who cannot sleep well at night.

... I do not need to sleep. I could be your guard outside the door, so I could wake you if something happens. Then you could feel safe?

[alethkars-true-king]

...You would do that? Won't you get bored of waiting around?

[story-cryptic]

You need your sleep! Humans get sick if they don't sleep enough.

I will think about the binomial theorem and your aunt's publications on Alethi history to keep myself entertained!

[alethkars-true-king]

I don't like having to sleep. I didn't have to sleep in the Spiritual realm. I didn't have to eat either. Eating and sleeping is such a waste of time.

Oh... uh... you do that, then.

[story-cryptic]

So many interesting lies... they were sweet and sad like yours, sometimes. Sometimes they were lemony!

I have been listening to other people! Ardents and scholars tell wonderful lies! But yours were deeper, and so I came.

[alethkars-true-king]

I am not lying! You're confused. You're picking it up from someone else.

Did you say scholars? What are the scholars lying about?

[story-cryptic]

MmmmMmmMmmmmhh... fascinating. Can humans lie without knowing they are lying? I think this is true. I might have watched Elhokar do it.

Scholars lie about many little things! Like scholar Rushu, who says 'I have only had three cups of coffee today' when it has been seven and three eighths! They also lie about respecting someone's academic achievements, when really they are thinking "this person is wrong and an idiot."

[anremithrl]

Denouement truly understands academia.

[anremithrl]

Someone asked me why you couldn't just pick the countable model of set theory with countable choice, and I was trying to explain the difference between internal and external countability, and the limitations of math and first order logic and I don't think I got it across and I don't know how to really explain it. Anyone got a good explanation/intuition for it?

[lilith-hazel-mathematics]

When a theory or model says "there exist such and such", it's kinda like saying "yeah I know a guy". When it says a certain object is countable, it's saying "there exists a function which bijects this set to the naturals", which is basically like saying "yeah I know a guy who counted everything in this set once". If a theory says a model of another theory is countable, that's kinda like saying "yeah I know a guy, Matt, and I know this other guy Fred that counted him". So you'll say Matt is countable, because you know a guy. But if Matt doesn't know Fred, then Matt doesn't know a guy, so Matt says he's uncountable. It doesn't matter to Matt that you know a guy, because Matt doesn't. So you say Matt is countable, and Matt says he isn't, and in a way you're both right. Countable models don't know the guy who counts them.

Fred is kinda like Yog-Sothoth, if you're into that kinda thing. He knows everything, everything about Matt in particular, but he's locked outside the universe. He's a Matt superfan locked outside Matt's house. Matt doesn't know that guy, and he'd probably go crazy if he did. Maybe because Fred is really fucking annoying. "Stop counting me!" Matt would say. "I'm just gonna diagonalize again!" he'd protest. "I'm calling the cops!!!" and he breaks down. But Fred keeps counting. "I know how many objects are in your house". Because he is that guy.

[lilith-hazel-mathematics]

To be a little less silly... You could also maybe could explain it by analogy to computable functions?

There are only countably many computable functions, because each computable function can be identified with the shortest (lexicographically) computer program which generates it. Computer programs are just text strings, and the set of text strings (over a finite alphabet) is countable, so that's it.

However, the bijection produced that way cannot be computable! If it were computable, then you could diagonalize to construct a computable function not in the enumeration, a computable function not listed in the list of all computable functions. Impossible. So although the set of all computable functions is countable, there's a precise sense in which the computable functions "don't know" they're countable. The bijection which witnesses countability is itself uncomputable. It exists outside their reality.

Similarly, countable models of set theory "don't know" they are countable. A similar diagonalization works here. Let M be a countable model of set theory, which obeys at least the axioms of Infinity and Specification. Since M is countable, there's a bijection F:ℕ→M. In pursuit of contradiction, suppose F were represented by a set inside M, or at least definable in M. Then, letting S = {n∈ℕ : n∉F(n)}, we infer S∈M since F was definable in M and M obeys specification. It follows that there's k∈ℕ such that F(k)=S, because F was bijective. By definition of S, we have k∈S iff k∉F(k). Since S=F(k) by definition of k, then k∈S iff k∉S, a contradiction.

So if M really is countable, then the bijection F which verifies this fact cannot be definable in M. In other words, M "doesn't know" it's countable. It can't "see" the F which verifies the countability of M, so as far as M is concerned, M isn't countable. M being countable has no bearing on its internal properties, because internally speaking, it isn't countable.

[anremithrl]

Oooh! These are good. I tried to explain Lowenheim Skolem and how first order didn't have enough "information" to differentiate this well, and tried to explain with the set theory example and whatnot but erred on the side of too much formality. I tried explaining how the diagonalisation can't happen internally so you cannot "find" the missing elements and write down a lot of symbols only to get a very uncertain "I guess?". These analogies are good. I think the computable functions is a great analogy.

[theadhdgoblin]

[theadhdgoblin]

[anremithrl]

Someone asked me why you couldn't just pick the countable model of set theory with countable choice, and I was trying to explain the difference between internal and external countability, and the limitations of math and first order logic and I don't think I got it across and I don't know how to really explain it. Anyone got a good explanation/intuition for it?

[anremithrl]

Hi I like your blog :)

Also I'm not related to that anon who said the exact same thing. If there even is such a person. I'm just guessing.

Someone did in fact recently ask me that.

Also! I really like your blog tooo! It's kinda awesome.

Thank you so much!

[anremithrl]

Hi I like your blog :)

Thank you so much! You have made me smile!

[jennydolfen]

I love these moments when we're reminded that Kaladin is a teenager who canNOT keep his socks in one piece even though he's only owned them for a couple of weeks at most. He had no socks as a bridgeman. Boy went and ruined new socks in an eyeblink.

[official-cisphobe]

here's a random word generator--whatever word it gives you is now the thing you are the deity of

[nevergonnagiveyaup]

I am now the deity of Height.

I find this funny because I’m only 5 foot

@miapanktrot @notkiramoon505 @sage0clock @p3skybirdd @nerd-alerts @beanzdabean @itsbeeugaymers @ur-bestie57 @multishipper33 @onceuponastori @aussiebeth @noticethelotus

This is a tag game now

[multishipper33]

I am the Deity of Second. all bow down to your magnetic majesty, for I specialize in the silver of a consolation prize and a pocket watch.

@deltenner @genderless-spiderlily @callofwinter @strawbxrrypies @peskybirdonyx @aries0nmars + open

[strawbxrrypies]

I am the Deity of Opposition. I like this one. hehe.

@myphycopharmacologist @peskybirdonyx @miapanktrot @djyapyap33 @rzkx @ranchersghost @reguiarsalad +open tags

[reguiarsalad]

i am the deity of proportion, which is ironic as I suck at drawing proportionally

@itsezra @somethinsillyig @cattreats4humans @reyisodd @eudicotyledon + anyone else

[itsezra]

the diety of overcharge... except I have a problem where i always do too little rather than too much

@fern-does-stuff

[burrbeast]

I am the deity of Substitute

For when the other deities are off doing something else.

[anremithrl]

I am the deity of expansion.

I'm mostly used to expand space and make things bigger on the inside.

[nooneinnportant]

Princess x court jester except they're both technically knights?

[nooneinnportant]

Princess x jester except they're both technically knights, and the Princess is also a scholar who later becomes a queen, and they're both actually semi divine beings with the powers of a fragmented god.

Except the jester actually killed god, and he's also worshiped as a key figure in several major religions across the universe, and the queen/scholar/knight is also an atheist?