/sci/ - Science & Math

i guess it has something to do with the obvious contradiction

>>9557340
What would be the usefulness of that?

>>9557389
not OP but it would be interesting if you had some set that contained a contradiction or not depending on some other variables, kinda like an imaginary set. but im literally retarded so don't mind me.

>>9557340
You can obviously define it. That's what you just did in your post.

>>9557340
It would be the empty set. Any set that only has elements that are not inside it has to be empty.

>>9557466
The set above can be easily shown to not exist.

>>9557467
Yeah you're right, my mistake.

>>9557466
You can't define the empty set that way, because we know that x isn't in the empty set, which means that x is indeed in the empty set, which is a contradiction.

>>9557340
That's not how the axiom schema of specification works.

>>9557340
I thought about creating quantum recursive sets, where set definitions that depend on the set itself are defined as the limit of a sequence if it exists, which would naturally give the same result as normal set notation, but for set definition sequences which do not converge you end up with a 'quantum set' which is the sum of two or more sets. It works, but I cant find a single case where it gives any interesting or useful result. There are some problems which lead to multiple possible sets, but in general there turns out to be 2 'extreme' situations, in the case of OPs set you get A) the superposition of the empty set and the set of all elements in the universe you are working in S = {}&U, and B) the superposition of all subsets of the universe S = &P(U), where I use & as the superposition operator.

>>9557340
You can in New Foundations.

ZF set theory (and HoTT etc.) don't let you take the absolute complement of a set/type. Probably one of their more glaring deficiencies.

>>9558279
How the fuck do I into HoTT? Do I really need to be familiar with alg top and cat theory before?

>>9558286
You should know some category theory and general MLTT, and at least the idea of the fundamental group(oid) (which is pretty intuitive). General MLTT is most important though.

>>9558279
"absolute complements" never actually come up in actual mathematical practice. in fact, quantification over "all sets" is itself a huge meme

>>9558294
I was told the introduction to type theory was pretty good though, they made it seem like I could just use it and not have to read specifically about MLTT. By the way, is there any good paper that highlights the differences of type theory and "normal logic"? It's not clear to me what their fundamental differences are, for example, what's the difference between intuitionistic logic (where the notion of truth is proof theoretic) and type theory?

>>9558298
>the introduction to type theory

You mean in the HoTT book? I would combine it with Martin-Loef's original paper which is extremely clear (although it uses a few weird definitions/concepts).

>It's not clear to me what their fundamental differences are, for example, what's the difference between intuitionistic logic (where the notion of truth is proof theoretic) and type theory?

Type theory *is* intuitionistic. It's a refinement of natural deduction to include terms as well as types. Read ML's paper and this will all be clear.

>>9558296
>>9558296
>>"absolute complements" never actually come up in actual mathematical practice

Yeah but they come up in real life. You can say Gandhi is not a spoon without specifying any common set that they belong to, because they are both things that exist. This is really a problem of not having a universal set more than anything.

>>9558302
Just to be sure, you're referring to the paper from 1972 "An Intuitionistic Theory of Types"?

>>9558331
Intuititionist type theory (1986)

>>9557340
Can we define a set this way: [math]S = \{x|x \in S\}[/math] ?

>>9558345
non-empty set?

>>9558279

what's the point of allowing a nonsensical circular formula like OPs?

another stupid question, what do you mean by set complements aren't allowed? isn't the usual undergraduate set theory based on ZFC?

>>9558349
Nope, it might still be empty

>>9558345
This is equivalent of saying: A set with it's elements exists such that it contains its elements. This is true for all possible sets so it's kinda like dividing by 0.

>>9558345
no

>>9558345
Yes, you just did. Then you can easily show that no such set exists.

>>9559480
Except that every set fulfills this definition

>>9559480
in what way does that define a set anymore than saying "let S be a set"?

>>9559503
This makes no sense whatsoever.
>>9559509
"let S be a set" used in a proof is quantifying over every set. How is this similar in any way to giving a definition of a specific set?

>>9559480
Every set does fulfill this though. It's pretty much saying S = S though, so it doesn't really tell you anything about the set.

>>9559711
>Every set does fulfill this though.
This sentence makes no sense in context.

>Why can't I define a set like S = { something referencing S }
probably because doesn't follow from any of the axioms you uncritical dipshits

- in zf you need a superset to draw your elements from first if you want to use specification
- in nbg your bullshit attempt at projecting from some clever tuple will either give you a proper class or be empty
- in nf(u) you can finally come to terms with tbe fact that you need S to exist before you can use it in any (((stratified formula)))

>>9557340
>he's wondering why someone would use a predicate before defining it inside of a theory
You're a large set

>>9559936
for V

>>9558144
>superposition operator

>>9558345
You're saying S = S

>>9557340
In ZF set theory, the axiom schema of specification is limited to extracting elements defined by a predicate from an *existing* set (i.e. yielding a predicate-defined subset of the original set). You can't extract from the class of *all* sets, because it's a proper class and not a set.

>>9560674
Every proper class is trivially a set.

>>9560669
>He complains about the names of operators

>>9557340
Something something Gödel Escher Bach

>>9557340
We can. There's Zermelo-Fraenkel axiomatic set theory which includes the existence of