Set Theory

>>7955331
set theory is used by physicists. When they say that ''the mass is 10 kg'' they literally believe that there is a number 10 floating in space and somehow giving you the answer on your measuring device..

Now, what framework for numbers ? it is set theory, where numbers are encoded as sets.

Since physicists believe that the mathematical inferences describe the world, when the inferences lead to quantified/numeric facts, they equally believe that the axioms of the set theory is true.
The first axiom of set theory is ''there exists a set''. so physicists believe that there is some set floating around.

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>>7955331
>using \subseteq instead of \subset

>>7955331
What are you still doing here? I thought I told you to go fuck your mother.

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>>7955337

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>>7955347
Go home and get your fuckin shine box

>>7955331

math grad who subsequently worked in a mom-and-pop, not-modern factory (recession etc). I stuck around long enough to understand certain details of the supply chain, and common parts in products. I literally set myself a basic (naive) set theoretic exercise to explain the plant's various materials and their flows, completing my treatment of such shortly before leaving.

>>7955331
Finite sets are used all the time. People use datatypes representing finite sets in mundane programming work.

Infinite sets are used all the time in mathematics. But if you just want your formula sheets, you may not need to care about them. You'll never see one in real life.

ZFC sets are wankery, and the meme of using them as a foundation will be dead within the century.

>>7955331


I am currently reading Ian Hacking’s Why is there Philosophy of Mathematics at all, and it is mostly about the contemporary dabate platonism/ nominalism, so I would recommend it as a good place to look for an answer to this question. A crude copy-paste is given below.

Hacking asserts that it is Paul Bernays who introduced the modern idea of 'platonism' and further he writes about the different brands of platonism, noting (p228):

After Bernays had introduced the word 'platonism' to the philosophy of mathematics, it should have been clear that one should not speak of platonism, but about platonism restricted to some domain of objects, such as the class of integers, or any Zermelo-Fraenkel set, or whatever. Perhaps even something like the class of all ZF sets, in a van Neumann-Godel-Bernays set theory.

So a good point to remember is that according to him

Absolute platonism, asserting the existence of all definable mathematical objects and relations, is untenable. What remain are relative platonisms. The weakest interesting platonism described by Bernays asserts the existence of a totality of whole numbers...a twenty-first-century platonist will say: positive integers are abstract objects. Kronecker said that God created them. So he thought they exist. That's a species of platonism - about numbers. He was a very modest platonist, but a platonist all the same.

Hacking comments:

Such a thought seems never to have crossed Bernays' mind, for he thought of platonism in terms of totalities rather than 'abstract objects'.

So Bernays turns out to be a restricted platonist while

In Bernays' vocabulary Boolos is a cautious platonist. He has no problem about the totality of whole numbers, but he has many qualms about sets whose existence is proven within Zermelo-Fraenkel set theory with the axiom of choice.(229)

Perhaps there is something of an answer in this heap of quotes

Physicists use topology, Analysis, Algebra, Differential Geometry, probability theory, etc. and all their offshoots l like calculus, Differential equations, elementary algebra, Linear algebra. Set theory is the rigorous foundation for all of this.

>>7955331
Consider the set of things that can't be described by set theory. This set is empty. A proof by contradiction is as follows, suppose it is not empty, then consider any object in the set. By virtue of being in the set it must be describable by set theory and therefore not in the set, hence a contradiction. Thus the set must be empty.

It follows from this that you can use set theory for everything. OP, you are clearly brain damaged if you didn't notice such a trivial observation.

>>7955345
>using \geq instead of meme arrows.

This is you. This is what you sound like.

>>7956053
retard

>>7956050
Fatality

Flawless victory

>>7955331
>when is X used in the """""""""real world"""""""""?

Can we make these kinds of threads a bannable offense?

>>7957618
russel's paradox?

>>7955337
this sounds like a plausible interpretation in the philosophy of mathematics

>>7956050
10/10

>>7955331
When is lifting dumbells used in the real world?

How many set theories are there in 2016 ?

>>7957746
When gettin' pussy.

>>7958387
he lives to be noticed by a few girls

>>7958417
Let's be honest nobody would bother with pure maths if they didn't have the chance of being noticed by a top professor.

>>7955345
\subseteq is a linear order

Linear orders are generally more useful in maths than strict orders.

>>7958767
It isn't. Just a partial order. Moreover, the choice is purely aesthetical here. \subseteq have exactly the same meaning as \subset. The strict relation corresponding to them is \subsetneq.

Set theory is used all the time in probability theory and by extension statistics, which are incredibly important in all branches of science, policy, and sometimes business.

>>7958816
>mfw trying to explain set theory to statistics students when they're going over beginner probability

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>>7955331

The Cycle of Rain

When is this shit actually used in the real world?

That's what you think you're 'learning' at school, anon? What a fucking pleb.

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>>7955337
>it is set theory, where numbers are encoded as sets.

Which is a construction nobody uses since logicism died almost 100 years ago.

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>>7956050
Flawless logic

>>7955331
It's basically THE foundation for topology.

>>7955331
I like that guy

>>7955977
Is it though? How much do you really need ZF/cumulative hierarchy/new foundations/whatever set theory to do any of that?

Differential geometry is often formalized through category theory, algebra can be done equally well with logic or category theory, and in linear algebra I really only remember it coming up with reference to needing choice for infinite bases.

>>7958986
also, point set topology is bland as shit. Even poincare was against the set-theoretic treatment, and you will find that set-theoretic issues in topology are comparatively rare, really only when it comes to size. And of course, with the theory of locales, you can do quite a bit of topology without set theory.

I'm not saying its useless, but most math is studied "synthetically," more or less meaning that the underlying set theory almost never matters, and in many cases this can be made concrete by switching foundations, with most results remaining constant.

>>7955361
I would really say this is the most accurate description; its just a data type like any other. If you in particular need to be able to recursively construct infinite sets in this way, its useful, but otherwise, there are replacements which work just as well or better for most situations.

So, >>7955331, I think its really useful just in being a naive foundation to get off the ground if you are worried about that sort of thing, and it has some nice (but also some not very nice) properties which are useful to abstract from (or see as examples of common abstract structure). It's not really a "real world" issue, except maybe in the case of needing to use it as a nice "data type" for very specific situations, which require an ontology like set theory (which btw is quite a weird one).

Set theory doesn't seem to lead to any particular solutions to questions other than size really (making formal that there are multiple infinities, that there is no largest infinity, how properties of models change when going between infinities).

>>7958977
tell me about it; I'm stuck in a shit field that looks up to this kind of thing and thinks it's the state of the art, end all way to approach formal science and it is fucking retarded

Occam razor bullshit that has nothing to do with reality, usually couched in "sets are the simplest structure because they use one symbol".

>>7958816
>is used
and
>by extension
don't go together

A Javascript developer doesn't know about fibre cable physics or even about the CPU.

The mathematical scenario is even weaker than that, because foundations like set theory (even if it's just sigma algebras you define in terms of them) are not even necessary to reason (more or less formally) about the front end theory, probabilities of things.

>>7958977
But what do scientists use then ?

>>7955977
>Set theory is the rigorous foundation for all of this.
I keep hearing this but am too dumb/not enough time to figure it out myself. Is there a good book on it? I'm slowly working through spivak now.

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>>7956050

>>7959595
That's a good start, though if you want to learn how set theory is the foundation, read up on the ZFC axioms

>>7959606
Thanks bud!

>>7958977
By that, you mean the dominant method by which modern mathematicians formalize math. You have an odd choice with words.

>>7956050

I think you made the mistake of assuming that the set of things that can't be described with set theory is actually a set... why is it a set in the first place?

>>7960116
Do you have a proof otherwise?

>>7956050
>>7960116
>>7960526

The Axiom of Specification restricts the creation of " the set of things that can't be described by set theory" if your doing this in ZFC.

>>7955381
>as no problem about the totality of whole numbers, but he has many qualms about sets whose existence is proven within Zermelo-Fraenkel set theory with the axiom of choice.
I find this so silly. It seems hypocritical. The axiom of constructibility implies choice. So those who deny choice believe there are non-constructible sets. This seems much more unreasonable than choice itself.

>>7960543
That's true and all, but the Axiom of Constructability isn't part of the Zermelo-Fraenkel axioms. Its an added Axiom instead of the AoC.

>>7960543
>So those who deny choice believe there are non-constructible sets
is this valid only in classical meta-logic ?

>>7960549
Well, if you deny AC, you must believe there are non-constructible sets, which seems less reasonable than AC itself.

>>7960555
I don't know quite what you mean by that, but from ZF + Constructibility there are not only choice functions, but a definable class-function that well-orders the entire universe.

If you're trying to ask if it's constructively true, then yes.

And it seems only reasonable that a constructivist should believe that every set is constructible. Therefore a constructivist should believe in the axiom of choice.

in the gym

/thread

>>7960541
>working in ZFC
lol

>>7961871
As opposed to what? ZF?

If you reject ZF, please specify which axiom you reject. If it's any but regularity, good luck doing any type of math.

>>7958387
True. But its not the lifting of the dumbell that gets you pussy. Its how toned and developed your muscles are.

>>7961927
Not her, but the axioms of ZFC are statements about sets whose members are themselves sets, and so on all the way down. It is not at all necessary to convert every mathematical object into such a weird type of set. We can still work with things like sets of integers or sets of real numbers, without treating the integers or real numbers as sets themselves. So it's not necessarily a question of rejecting certain axioms of ZFC. They simply aren't relevant. ZFC is pig disgusting not because of the choice of axioms, but because of the choice to have only one type of object.

>>7959339
There are lots of statistical problems that are more sophisticated than finding a mean and standard deviation.

>>7960649
>>Well, if you deny AC, you must believe there are non-constructible sets,
can explain why ?

all we know is that The axiom of constructibility implies choice. so how do you go from this implication to yours ?

>>7955331
there are people in the real world that believe birds and animals are two distinct sets with no common membership between these two groups.

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Intuitionistic type theory is objectively superior to set theory. Prove me wrong.
Protip: You can't.

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>>7962665
>Intuitionistic type theory
This is just a formalization of kant's doctrine.

>>7962264
bump

>>7962154
Guess you don't know of set theory with urelements then.

Anyway, set theory with many types of objects can certainly be done. It's just that it's silly because any model thereof can be embedded in a larger universe in which the only objects are sets.

>>7962264
Do you not see how it's the contrapositive? "Every set is constructible" implies AC, therefore the negation of AC implies not every set is constructible.

>>7963725
>types are silly
>[math]3 \cup \geq[/math] is not

>>7963726
I edited my comment to remove my response to intuitionistic type theory because it is too silly to even talk about.

>>7963731
Not nearly as silly as coercing everything into untyped sets.

As for intuitionistic logic, classical logic can be embedded in it. You can do classical logic inside it if you want, but it extends classical logic with all sorts of useful connectives, like a version of "A or B" where the proof not only shows that A or B is true, but which one it is. This deepens the Curry-Howard connection.

>>7963739
I have only a cursory understanding of intuitionism, understanding its axioms and notion of truth/proof, and I dismissed it quite quickly as silly. But if classical logic can be elegantly embedded in, whence as you say it can be view as an extension thereof, maybe it isn't silly after all. I'll have to look more into it. Thanks.

>>7963725
>>Do you not see how it's the contrapositive?
okay, so you do believe that the contrapositive is equivalent to the initial implication.

>>7964232
Ah, you don't believe in the logical equivalence of the contrapositive.

What I said is mathematically true. By rejecting the equivalence of contraposition, you reject mathematical truths. There is no place to go from here.

>>7959287

Try measure theory without sets m8.