What's your favorite thing about set theory, Veeky Forums?

>like some kind of math deity

never gets old

>I particularly like how sets can be used to define functions.

What do you mean? Functions are sets you dummy.

Which are those more normal ones?

I stopped caring about math when I was introduced to the concept of real numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't even be represented with fractions, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

I stopped caring about math when I was introduced to the concept of rational numbers. What a crock of shit. If your equation can only be solved by inventing numbers that have imaginary "fractional parts", like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

I stopped caring about math when I was introduced to the concept of integers. What a crock of shit. If your equation can only be solved by inventing numbers that have a magical dash sign to the left, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

Nostalgia

I'm not this guy, but if you're implying that the function
[math] f: {a,b,c}\to {d,e} [/math]
given by
[math] f(a):=e [/math]
[math] f(b):=d [/math]
[math] f(c):=e [/math]

is nicely modeled as set of pairs

[math] \{(a,e), (b,d), (c,e)\} [/math]
or even
[math] \{ \{ \{ a \} , \{ a , e \} \}, \{ \{ b \} , \{ b , d \} \}, \{ \{ c \} , \{ c , e \}\} \} [/math]
then you're just indoctrinated.
"functions are sets" is fairly obscure as it doesn't even capture algorithmic aspects, intensionality, evaluation time, things like runtime. It can be useful, but it's generally a brutal thing to do.

I stopped caring about math when I was introduced to the concept of natural numbers. What a crock of shit. If your equation can only be solved by inventing numbers, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

The set of definable reals is not a definable set.

Moreover given any definable infinite set of reals you can't define the subset of definable reals within it.

I stopped caring about math when I was introduced to the concept of logical deduction. What a crock of shit. If your equation can only be solved by inventing rules of deduction like "modus ponens", like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

[math]\{(a,e), (b,d), (c,e)\}[/math]

is correct.

Algorithms have nothing to do with mathematical functions.

Intensionality, evaluation time, runtime, etc. also have nothing to do with functions. What the fuck are you talking about?

What is a function if not a set of pairs that are related by the function.

f(x) = y if and only if xRy if and only if (x,y) is an element of R sounds pretty good to me.

I'm not this guy but I agree.

Another thing you lose out on when you define sets this way is that you don't have the domain and codomain. There is a better (imo) definition of functions in set theory where they are defined as triples including the domain, the codomain, and the ordinary function itself.

What this user is referring to is the computational notion of a function that's commonly used in computer science. Here functions may have internal structure which is sensible considering you can write two programs that accomplish the same thing in entirely different ways. This is what one means by intensionality (two functions are intensionally equal if they have the same internal structure and extensionally equal if they give the same outputs on the same inputs). The rest of the things referenced fall out of this internal structure.

Functions of this sort are typically formalized in some sort of type theory (eg. lambda calculus) or by other programming language related methods (eg. recursive function theory). They can also be modeled within category theory and they have a very close relationship to formal logic.

In Set Theory one only has extensionality but not intensionality.

Set functions are the type of functions in set theory. They aren't the only type of functions out there.

>using sets to define functions
Terrible, what if you want a proper class as the domain?

>algorithmic aspects, intensionality, evaluation time, things like runtime
None of these have anything at all to do with the theory of functions.

>Set functions are the type of functions in set theory. They aren't the only type of functions out there.

So your argument is "if we're talking about something completely different that has almost nothing to do with the subject at hand, then things work out differently." Fucking ace observation.

>muh NBK meme.
I wouldn't mind these sorts of comments if the people who posted them actually knew NBK and weren't just a bunch of algebraic geometry memers who don't really know much set theory beyond the basics.

>being this pleb
see

Then you have a class function, which is defined by a first-order formula. Class functions come up all the time in set theory at a fundamental level (such as the replacement axiom) and are trivial to work with.

Functions are a very ubiquitous concept that lie at the core of many different theories. Saying functions are sets and justifying it by arguing that the topic is set theory is no better than saying water is radioactive and justifying it by saying you live in a nuclear wasteland.

It's only true in a way no one cares about but it's obviously false in general.

>These people only like set theory so that they can have something to point at when people ask about rigor. They don't actually care or understand rigor.

I'd agree with this.

>AC makes ZFC inconsistent
It is an early result of Gödel that [math] \text{Con(ZF)} \rightarrow \text{Con(ZFC)} [/math]. In other words, if you believe ZF is consistent, you believe ZFC is consistent.

>the subject at hand
The subject at hand is "functions" and not "the notion of functions in set theory"

It's one way to capture the idea of functions, and not the only one.
Besides, the mere list of points doesn't even capture the codomain, merely the range. Even set theorists will sometimes define a function to be a tripple (f,X,Y), where f:X->Y and range(f) is a subset of Y.
Moreover, in your framework, are pairs a primitive notion? That's hardly ever done, the definition
(a,b) := {{a}, {a,b}}
is the standard one.
en.wikipedia.org/wiki/Ordered_pair

And yes, intensionality has a lot to do with mathematical functions. The expression "f(x):=10·x+2" and the expression "f(x):=2·(5·x+1)" are extensionaly equal functions. Are you gonna say people who care about
en.wikipedia.org/wiki/Confluence_(abstract_rewriting)
don't do math?

Maybe you've choosen a comfy dogma for you, but it's not the end of things.

Functions are not even around as long as one would think.

The problem I have is very pedantic. Enough that I can't find anything about it in the literature.

1) First consider that if I take a formal language over a finite alphabet then the set of all words in the language is countable (first we have the words of length 1, then the words of length 2, etc.. this yields an order).
2) Consider that ZFC is formalized on top of formal logic which is in turn formalized over a formal language over a finite alphabet.
3) As a result the set of sentences we can define in ZFC is countably infinite.
4) Once we give the construction of the reals in ZFC (or if instead we consider the reals as a model of ZFC) we have the result that every real is a set and there are uncountably many reals but only a countable number can actually be defined. This is a well known result as it yields the existence of undefinable reals.
en.wikipedia.org/wiki/Definable_real_number#Definability_in_models_of_ZFC
5) If one can produce a definable set containing only undefinable reals then the axiom of choice would fail on that set, one would be unable to define functions from/to that set, disproving the claim that the set is empty would be non-trivial, etc..

As a side effect of (5) you also can't take a definable set containing undefinable reals and produce a subset containing only the definable part.

Is there any compelling use for untyped sets besides that awful construction of the naturals as sets of lesser naturals?

Like Kim Kardashian, it's popular for being popular.

I'm so sick of set theory and modern math in general. It's complete mental masturbation and I'm sick of having to listen to pretentious idiots (muh grothendieck, muh serre representations, muh weil conjectures) who can't do real math (i.e. give original solutions to actually difficult problems, as opposed to regurgitating their textbook verbatim, or even worse, spewing incoherent gibberish that "sounds like" math (read: postmodernists)) say "math isn't about calculations, it's about ideas".

This people piss me off so fucking much. They're not good at math and they've fucking taken over the math departments. I have no idea how anyone can possibly fucking stand going to university these days. People think it was just the humanities that was ruined by "continental" philosophy, when really the math departments got a good dose of it too.

Has anyone seen the IMO problems of the past 10-20 years? They're a fucking joke compared to what they had to solve in the 60s. Math has become usurped by linguistics, pedants and crypto-theologians and it's driving me fucking crazy.

Your point (4) is a common misunderstanding. It is possible for each of the uncountably many reals to be definable without parameters, even though there are only countably many parameter-free formulas.

The consistency of this apparent paradox is subtle yet simple, involving Tarski's undefinability of truth. It is this: the correspondence between parameter-free formulas and the reals they define is itself undefinable (exercise).

In fact, Hamkins proved that it is consistent for every one of the class-many sets to be definable.

jdh.hamkins.org/pointwisedefinablemodelsofsettheory/

That said, even if we worked in a model in which there is a definable set of undefinable reals, there could still be a choice function, and we could still define functions to/from that set: just not in a parameter-free way.

I'm having trouble following the reasoning here. Could you elaborate some more? Are you using a double negation?

By definition a set if countable iff there exists a bijection between it and, say, [math]\mathbb{N}[/math]. The reason that it is possible for each of the uncountably many reals to be definable even though there are only countably many parameter-free formulas is that there may be no bijection between collection of definable reals and the set of parameter-free first-order formulas. I clarified that the obvious correspondence that simply maps each parameter-free formula to the real it defines is, in fact, undefinable (and thus is neither a set nor class-function, i.e simply does not exist in that universe of mathematics).

Uncountably many definable reals yet countably many definitions.

>you can construct uncountable sets without using inconsistent sets of statements

this is a contradiction. this is also the reason why no one likes set theory.

I can totally agree that the map from the set of formulas to the set of definable reals would be undefinable. I mean that just seems entirely obvious. What I'm not seeing is why that allows you to conclude that all reals must therefore be definable.

The flow of logic I'm seeing is like

>Suppose it is not the case that all reals are definable.
>???
>However since there is no definable bijection from the definable reals to the formulas then ???
>???
>Thus we have a contradiction and "it is not the case that all reals are definable".
>Then by using double negation elimination (non-constructive proof technique) we have that all reals must be definable.
or something

>>Thus we have a contradiction and "it is not the case that all reals are definable" is false.
fixed, typo

This is why I stayed away from mathematics higher than PDEs because everything after that is autistic inapplicable shit. I only care about math that has actual applications in science, the rest just seems like philosopher shit to me and with attempts to make math more "artistic".

So I don't need topology and calculus on manifolds to understand physics?

Pretty sure that is also copypasta and obvious bait.

No, you need some common sense. Topology and "calculus on manifolds" are just tools which you use to understand the world. In the same way that the names of things are not the things themselves (much to the discomfort of theologians and schizophrenics who wish to think otherwise), the tools we use to understand the world are also not the world itself.

Don't get so fixated on the formalism. I see this happen so often with the "good" students. They develop a system for getting the highest grade, but often at the cost of becoming detached from reality, and mistaking the formalism they are using to consistently get correct answers on tests, for reality itself. It's extremely frustrating try to convince them otherwise because they're so caught up in language that they quite literally cannot conceive of anything outside of the formalism. They have no intuition.

Not copypasta, not obvious bait, just severely disillusioned and frustrated by the current state of the epistemology of modern mathematics. The navel-gazers are winning and it is becoming the end of me.

>Q: what is set theory?
>A: a first-order theory with a single binary relation symbol
>Q: well then, what are the semantics of first-order logic?
>A: [set-theoretic definition of a model]
you can't make this shit up

It's not that even that it's inapplicable, we don't know whether a given formalism is inapplicable or not until it is tested, it's that the people who are creating these formalisms have ZERO concern for whether or not it is applicable, and thus are simply creating "aesthetic" math instead of useful math.

The worst part is that these people DEMAND to be recognized for how intelligent they are, and then DEMAND a privileged victim status when criticized for disrupting and not contributing anything to society.

Actually, wait a second, a group of people that creates division among their host groups through subversion only to claim victim status when retaliated against for ruining everything... that sounds familiar!

A formalism gives you a rigorous method of devising proofs and producing theorems.
Models are examples of things that the system can talk about.
There are things that may be true in a given model that aren't true in the formal system itself. In other words, by relying purely on intuition one may accidentally assume certain statements that cannot be proven as they are not always true (i.e. there are different models where those statements are false, intuitively even).

As soon as you decide to discard the formal system and just work within a model (i.e. your intuitive understanding of a thing) you lose the ability to make rigorous arguments and can become subject to paradoxes and contradictions.

In other words, you plebs should just drop out into science or engineering already. We don't need you wasting our time in math.

This bothers me too.

You're still stuck on the terms. I could rename all of the numbers to scribbles and logic would still work. You need to work on your ability to make analogies. Also, try to not be so afraid of speaking informally. People aren't computers, we can understand what you're saying even if it won't compile.

What you're saying can be summarized informally by the following statement.

>Some things can happens given one set of rules, and other things cannot happen in other sets of rules

When I explain FOL to people who have never heard of it or have trouble understanding, this is how I explain it. It never fails to work. The only people who "can't get it" are usually the pedants who like to role-play as living, breathing computers. Anyway, think about games. Anyone who can tell the difference between 2 or more games, and whether or not certain moves in certain games are allowed in other games can understand the relation between theories, models and interpretations. You can think of anything with rules as a game if you want. You don't need a graduate course in logic to understand this. You need a graduate course in logic so you can get cozy with faculty so they will let you co-author with them. You don't go to school to learn rules (you should know them before you get there), you go to school to network.

>In other words, you plebs should just drop out into science or engineering already. We don't need you wasting our time in math.

And again, this is where the 'pretentious' in 'pretentious idiot' comes in. Do you really think calling people 'plebs' is a good way to recruit people and get them interested in your subject? Why not be more diplomatic? What do you have to lose by opening up to people? Your precious precision of language? Afraid people will think you're stupid? What is it?

>>Some things can happens given one set of rules, and other things cannot happen in other sets of rules
No, you misunderstood. Same set of rules. Same axioms. Many different things can be described.

For instance, consider Euclid's axioms for Euclidean geometry. In this context consider the sentence
>If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.
Now, many like perhaps yourself, who think of Euclidean geometry as this totally natural obvious thing that we can all just picture in our heads without very much effort will find this statement obvious. However, as it turns out this statement is unprovable under Euclid's axioms.

In fact, one can devise a ridiculous example of "Euclidean Geometry" (that still satisfies Euclid's axioms) where that statement is false. However, if you had not realized that this statement was unprovable then you would have never even thought to look.

A more down-to-Earth example of this phenomenon is the birth of non-Euclidean geometries. They arose because for a long time people believed that Euclid's parallel postulate was "so obviously true" that it had to arise out of the other axioms. It wasn't until someone realized that by removing the parallel postulate from the axiomatic system (thereby obtaining incidence geometry) one can start considering other sorts of geometries where the parallel postulate is false (and different variants of it are true).

Your FOL example is okay but with that super-intuitive approach to FOL you will never be able to break out into intuitionistic or other logics (because they require a shift in perspective and semantics, i.e. they require different intuition).

Wow you really knocked this one out of the park.

I think you just set a double high-score for condescending and clueless.

I've *actually* read and worked through the entirety of Euclid's Elements alongside Hilbert's Foundations of Geometry and Carnap's Symbolic Logic as supplementary texts, unlike someone else who is simply plagiarizing the preface of every single introduction to Hyperbolic Geometry textbook/youtube video I've read, seen or heard of- I mean, c'mon man, you're literally quoting the COPY of the text, not even the interesting stuff.

also
>he think I've never worked within an intuitionistic framework
>mfw

This is too good. You really think know nothing about intuitionistic logic? How did you arrive at that conclusion? Do you judge every book by it's cover? Again, are you really afraid of people who speak informally? Is it too uncomfortable for you to conceive of someone who is capable of both thinking and writing formal proofs but also have crude and informal conversations about formal subjects on the side? Have you really never met anyone like that before?

This is probably the most autistic thing I've ever read on the Internet.

>hyperbolic geometry
>having anything to do with Pasch.

I only mentioned the parallel postulate as a babby example that even a babby like you could wrap their head around.

Hilbert gives a different axiomatization of Euclidean geometry (and he is far from alone in that respect).

>Fails to understand basic arguments and prefers speaking informally.
>Is confused that others would assume he's not familiar with intuitionistic logic.

Seriously, I don't even know how someone could have gotten
>Some things can happens given one set of rules, and other things cannot happen in other sets of rules
out of a discussion about models.

someone post the original pasta and its source

No.

I did not say that all reals are definable. I said only that it is consistent that all reals to be definable.

Math is a load of useless shit after highschool.

>Maybe the only backdraw to set theory is that you can only define a countable number of sets

This is true about any axiomatic theory.

>"math isn't about calculations, it's about ideas"

There's nothing wrong with wanting to do calculations. But if you're looking for understanding, then calculations aren't as relevant. The worst is when you do calculations that are neither applicable to real-world problems nor help to understand math better. A lot of algebraic topology, as well as old competition problems seem like this to me. Like who the hell cares that 10-page random formula X is less than 10-page random formula Y?

>A formalism gives you a rigorous method of devising proofs and producing theorems.
>Models are examples of things that the system can talk about.
>There are things that may be true in a given model that aren't true in the formal system itself. In other words, by relying purely on intuition one may accidentally assume certain statements that cannot be proven as they are not always true (i.e. there are different models where those statements are false, intuitively even).

Solution: add more axioms until the theory *is* intuitive :)

The set that contains all sets that don't contain themselves.

>The reason ((that it is possible for each of the uncountably many reals to be definable even though there are only countably many parameter-free formulas)) is that there may be no bijection between collection of definable reals and the set of parameter-free first-order formulas
I don't get this. Of course, for any real like \pi there may be many formulas specifying it, and thus of course there is no bijection.

What I think is happening here is that "definable" becomes formalized and thus gets to be something that has models and sematics and stops being something that means what we'd want it to mean.
I'm not convinced of an argument that goes "it doesn't hold scrutiny, because once we formalize it in our language (that we'll not doubt under and circumastances) then it can be seen that the concept isn't categorical and thus allows for worlds where the statement doesn't hold"

All sets contain themselves so that is just a fancy way of saying 'The empty set'.

>The reason ((that it is possible for each of the uncountably many reals to be definable even though there are only countably many parameter-free formulas)) is that there may be no bijection between collection of definable reals and the set of parameter-free first-order formulas
I take it you say this non-existence (of thing that would would witness that the number of formulas is uncountable) opens up for a model of reals that's just not really "uncountable" from some perspective. So it's something not really real-number'ish

What I think is happening here in that reference (a geuss of me) is that "definable" becomes formalized and thus gets to be something that has models and sematics and stops being something that means what we'd want it to mean.
I'm not convinced of an argument that goes "it doesn't hold scrutiny, because once we formalize it in our language (that we'll not doubt under and circumastances) then it can be seen that the concept isn't categorical and thus allows for worlds where the statement doesn't hold"

It's been a while, since i lost this hard.

Doesn't bother me because semantics to me is merely a tool for proving consistence of theorems.
Imho "what are the mathematical semantics" as opposed to real world applications that motivated the language" is a question that can well and legimately be ask long after the set theoretical framework was developed. I.e. we don't need semantics for the logic to make sense. This my perspective carries ideas of formalism, of course.

[math]\mathbb R[/math] is in fact itself a real number.

Yeah, and it contains itself. What is your point?

When defined as dedekind cut of what does this mean?

If we take Cauchy reals and a real r in R is a huge set of equivalent sequences, I doubt R (a set of such reals) is itself a real.

Penetration

My point is that [math]\mathbb R \not \in \mathbb R[/math] you moron.

x being an element of y is not the same as x being contained in y.

In the sense that you do not need to be an element of something to be contained in it.

Troll harder.

>x being an element of y is not the same as x being contained in y.
That's what the original guy was referring to when he said sets which don't contain themselves.

The set that contains all sets who have themselves as an element

is not the same as

The set of all sets that contain themselves.

Troll harder.

It's a common way that Russel's paradox is phrased. I don't know what you want me to tell you; its incredibly clear from context what he was trying to say.

What he was trying to say

is not the same as

What he said.

I understand what someone is trying to say, but he said it wrong

is not the same as

I'm going to pretend that I can't handle minor syntactic issues while parsing a simple thought and proceed to fling shit for a half an hour

Also, sets that contain themselves in the same he is speaking do not make sense.

{a}
{a,{a}}
{a,{a},{a,{a}}}
...
it will never contain itself.

Let A = {A}.

The set of all sets of size at least n. The set of all nonempty sets. You're not even trying.

Of course this causes problems, which was the whole point of the original post. Jesus.

It sounds to me that the problem is infinity.

Alert all the mathematics departments around the world, we are dropping infinity as a valid concept.

Well, good thing that was solved.

In cases like withe the standard set theory ZFC, sets are protected from containing themselves by adopting the regularity axiom:
∀X.[∃x.(x∈X)]⇒[∃(y∈X).∄(z∈X).(z∈y)]
"For any nonempty set X holds that at least one of its elements y is not sharing any elements with X".
The inconsistency of a potential self-containing set S={S,...} follows from observing that {S} doesn't fulfill the required axiom.

However, there are set theories which reject this
en.wikipedia.org/wiki/Non-well-founded_set_theory

And afaik the theory of the Yokuzuna of math deals with such odd stuff too

Understanding the foundations will help you with proofs later in life. The entire point of mathematics is that it's exact, and as long as you just make up your own shit and ignore set theory, you're never gonna get anywhere. If you create your own model depending entirely on your intuition, it's bound to be inconsistent. That's not gonna happen if you just develop it using sets.

my favorite thing about set theory is that someone else has done all the intricate work for me and i can just do my unions and intersections without thought worry or care

Theres too many qualifiers for me.
The regularity axiom implies
S={S} is not okay?
S={S,{}} is okay?

Oh, I understand now.

If we assume that the reals are definable then it doesn't lead to a contradiction. However there is no proof that the reals are definable.

Is it also consistent that the reals are not definable?

Shouldn't it affect any theory formalized in a language over a finite alphabet?

Instead of restricting our view to the things that already seem obvious to us. Why not expand our view and find intuition in the wild untamed madness?

The approach in mathematics is always to generalize whenever possible, so why should this be any different?

There exist notions of countable reals.

You should say "If we assume every real is definable, then etc," as the term "the reals" usually refers to the set [math]\mathbb{R}[/math], which is certainly definable.

And as I noted at the end of my first post, even if some reals are not definable and there is thus a set of reals each of which is not definable, there can certainly still be a choice function from e.g. [math]\mathbb{\omega}[/math] to that set — it just wouldn't be definable!

Definability is not at all a requisite for a set to exist, but it is consistent to believe that every set is definable, so you are welcome to require this of your mental model of the universe of sets.

Don't take offence but it seems you do not really understand axiomatic set theory or the utilized model-theoretic concepts. If you have foundational concerns you may be interested in studying these topics.

Since the axioms (of any set theory really) will let you prove that if {a,b,c,...} is a set, then so is {a,c,...}, you are still brought back to the inconsistency of {S} existing

here more on the axiom
en.wikipedia.org/wiki/Axiom_of_regularity

>Definability is not at all a requisite for a set to exist
This clears up everything. I was using definability as a requisite for existence but you are correct, we are really dealing with existence of objects within models of set theory.

So we can assert a standard model of set theory and just be okay with a large portion of the model not being definable.

You are not alone, my finitist brother. e.g. probability theory and infinites do not work together (two envelope paradox).
And arguments about absurd ideas like uncountable infinites (cantor vs kronecker) weren't fought in the fields of science, but were decided by popcultural infestation of cool-sounding shit on god and "infinites beyond the reach of descriptive science".