Set theory gives math a solid foundation

> tfw this was my set theory book
> tfw set theory is too easy for my university, so the professor gave pop quizzes and wouldn't put notes online
> tfw I dropped out of college to become a gangster because of this

Take a real set theory course. I took a course that was dual listed as a grad course in set theory.

let's say 5 years

I don't see the problem with this?
For other viewpoints to help you understand there's: and a lot of stackexchange answers to this question.
The left set has an element, the right doesn't, so there's no way that they can be equal.

And categories are set-theoretic retarded

So?

en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

And this is retarded because ? I mean there are many weird things about set theory but this is stuff a toddler can handle

Kill yourself

Last time someone mentioned set theory here I asked OP what is so fucking wrong about set theory. OP never cared to respond
For real, I don't understand. What is so bad about set theory? can anyone give me the reasoning behind that idea? The OP in the last thread just limited to quoting gauss on it saying "set theory is bad" -Gauss

axioms of ZFC are retarded. you can only see the retardation once you look at other axioms.

if anything, ur-elements must be the main set theory

t. set theorist

You said literally nothing

>axioms of ZFC are retarded. you can only see the retardation once you look at other axiom
wow thanks I can see the light now
retard

t. dumdum undergraduates who do not learn set theory and yet want to discuss it

>people not understanding the difference between the empty set and the set containing the empty set
>people not understanding what it means to be a subset of a set
>people not understanding what it means for something to be an element of a set

Give up, OP.
Be a ditch digger instead. The world needs those too.

As said, there's nothing inherently wrong with set theory and it's not that set theory is incapable of expressing all mathematical concepts. In fact that's exactly the problem: it's so general that it allows people to express all sorts of bullshit that is mathematically meaningless and bears zero resemblance to how mathematicians actually think.

To use a programming analogy, set theory is like machine code, manipulating bits called 'sets' that have no meaning on their own. Category/type theory, on the other hand, would be a bona fide programming language: a series of 'high level' instructions describing exactly what you care about with explicit disregard for how the implementation is to be carried out. As long as the instructions are identical, two different implementations will still be considered 'the same'.

they're right tho

there's a bunch of reasons, many aesthetic, but they add up.

one is simply that the element/subset distinction is kind of artificial. category theory gives lots of examples of structured things where "sections" or "global elements" do just fine to serve as elements (effectively 1-element subsets). the cumulative hierarchy is almost never used, and all that data is usually thrown out when doing real math anyway. talking about the "structure" of sets is actually quite annoying, as rarely is it the full inner embedded elementhood structure that matters. i.e. what does it even mean to define a set up to iso? something nasty about the model of ZF.

the inner language of category theory on the other hand abstracts away from this artifice. kan ext, adjoints, limits, etc. all lead directly to definitions you want often, and get rid of arbitrary set theoretic distinctions. E.g., think of the ordered pair. In set theory, you have to form it as {x,{x,y}} or {{x},{x,y}} and they all have different elementhood structure, which really shouldn't matter at all. In fact, you can't avoid having annoying elementhood structure, even though you just wanted to pair something; similar to the hierarchy in general. Part of this is the weird ontology of set theory, which has to assume some "universe" of "concrete things", which was what OP was complaining about, i.e. I have to fix a universe where I know what each object is, so that when I union {x} and {y} I get a two element set, but not {x} and {x} - this requires a rich ontology that is not based on mathematical structure, and something you probably want to avoid. Category theory does not work like this, you specify this all structurally, with no need for elements. A cartesian product is defined as a universal way to project out to two sets, an element is simply a map from a terminal object, and unions are formed as a pushout of two monomorphisms. Everything is relative, so doesn't rely on some weird underlying ontology.

lmao, where is this from?

I like to cite this one ugliness:
If you model the natural numbers and the ordered pair in the standard way
(0:={}, 1:={0}, ... see and (a,b):={{a},{a,b}}, see ),
you prove
[math] 1 \in (0,7) [/math]

Pretty much (). The disjoint union

{1,2}+{2,3} := {(1,L), (2,L), (2,R), {3, R}}

can at least be modeled within set theory just as well.
In the category of sets, there isn't even a distinction from the global elements "choose_2" and "choose_3" in Hom({0}, {2,3}).

This relates to (). We shouldn't confuse those notions, just because they are called sets and elements.

Afaik the normal union can be reintroduced in topos theory, but only for sets that are together contained in a bigger set, and in a pretty complicated way. Instead of just saying "well my global elements are the normal elements", the idea is to express [math] \in [/math] itself in the topos, and thus get a normal'ish set theory:
For a subset [math] S \subset X [/math], the proposition
[math] x \in S [/math]
is rewritten as
[math] (S,x) \in \varepsilon_A [/math]
So here we must have
[math] \varepsilon_A \subseteq {\mathcal P} (X) \times X [/math]
where the power set denotes an object in bijection with [math] \{0,1\}^X [/math].
To make this above [math] \subseteq [/math]-expression formal, we need the subobject classifier diagram.
The ingredients are allready there, though, because there is an arrow from [math] \{0,1\}^X \times X [/math] to
the subobject classifier [math] \{0,1\} [/math], namely the eval-functions. The pullback of !:{0}->{0,1} along eval gives us [math] \varepsilon_A [/math], whos global elements [math] (S,x) [/math] are now more coherrently interpreted as claims [math] x \in S [/math]

I disagree with the glorification of cats. Stuff like repeatedly jumping to the category of arrows into a thing (overcategory) and consider the arrows there to be triangles - nobody would think natively in those terms.

>If you model the natural numbers and the ordered pair in the standard way
and what kind of dumb cunt fucking assumes some concrete set-theoretic construction of the natural numbers in their arguments?

You can construct the reals via dedekind cuts or as equivalence classes of cauchy sequences, nobody gives a flying fuck. As soon as they're created you only work with their abstract properties, like with any mathematical object really.

If you're not already thinking modulo isomorphims, no different formulation of mathematical foundations will be able to save you from asking retarded questions.

2016 and still not doing POINTLESS TOPOLOGY

fuzzy sets are the best sets

prove me wrong

>the point
>your head

and then some category theoretical constructions on subsets S of X give us a subset of X that's the union as we'd have it in ZFC.
I don't know more, because we probably shouldn't even care.

Yes, you are exactly right!
We don't use the set theoretical model but, at best, the truths expressed in the Peano axioms.
This is an argument against set theory as foundations. That's the "sure you can model stuff, but it's unnatural and not used once the theory stands".
While the successor function S in set theory is modeled as taking a set n to a set
n union {n}
(and for the model of the natural numbers this then amounts of taking e.g. 3={0,1,2} to 4={0,1,2,3})
the topos theoretic setup mirrors the rules of arithmetic more directly. The successor function is a function that has the properties you need for what it's used for - induction.

But foundations fights are somewhat silly in general, imho.
Still, I'm not arguing for topoi in it's most abstract incarnation either.
(I'm for a small constructive base and am sceptical of the power set axiom for sets, but that's another story)

in case the labels are a bit abstract, I'd translate
z... initial_label (zero) (in N)
q... first_choice (in A)
s... next_label
u... A_enumeration
f... next_term_of_A

A natural number object in a topos is an labeling object, so that

>A_enumeration(initial_label) = first_choice
and for any label y in N
>A_enumeration(next_label(y)) = next_term_of_A(A_enumeration(y))

That's also what's done in dependent type theory.
In fact, they introduce a notion of set not very soon in the homotopy type theory book and the prove that the natural numbers defined that way form a set (just meaning the set operations on its global elements are well defined) isn't straight forward.

you do realize that that was the exact point of that person's post; set theory introduces a lot of annoying irrelevant bulk, and my post made your exact point already but applied it to the discussion at hand

>doing category theory instead of set theory

>not doing species theory through Shinichi Mochizuki inter-universal Teichmuller Theory

...

>Precise statements concerning such issues, however, lie beyond the scope of the present paper [as well as of the level of expertise of the author!].

He's so sweet

subset or element is a pretty major distinction to be made

It's a pushout you undergrad.

People would always get tripped up on this in my foundations of math class. I always thought it was fairly intuitive that the empty set is not the same thing as the set containing the empty set so long as you recognize that a set in itself is an object and the empty set is still a set.

90% of people are brick wall retarded and think fractions are evil witchcraft not even the bourgeoisie use and calculus and QM are subjects that needs a "leap" to understand

>OP doesn't get this

>dedekind reals

burger.jpg

what is the burgerman's problem with dedekind reals? they are a rigorous, well-defined (and on top of that, elegant) construction of the real numbers

actually, don't answer that, I don't really want to know

lol is he the terry (templeOS) of sci
made his own scizho math from scratch?

While it is possible to define some of the reals explicitly it is not possible to define all of the reals explicitly. This is an obvious result from formal language.

Burgers' problem is that if you're defining your your sets in raw set theory (i.e. logic) then you sometimes find yourself in situations where you have to give a sentence of infinite length in order to define a number.

I think Burger would accept that you can define a proper superset of the rationals this way but he would argue that it only gives a proper subset of the reals. Kind of like the algebraic numbers.

No, he created a new branch of math on top of existing theory but he has been developing it on his own for years and few people understand his work. The few that have been studying it however agree that it is legit (a few small mistakes were found that were corrected).

ye so that is just like Terry. his OS works and has no major flaws. but its completely useless, and he is insane. im assuming this guy is insane too

He proved the ABC conjecture.

If you can't even grasp set theory this basic you shouldn't be commenting on what should and shouldn't be the foundation of mathematics.
>these are the people giving you math advice on Veeky Forums

we will see about that

This complaint seems incoherent. Take the constructivist's real number line. The constructivist can still show by diagonalization that there are uncountably many reals. However, the only reals we can define in this setting are computable, so from our metatheoretic vantage point, there are only countably many reals.

So why isn't there a problem? Because the uncountability of the reals is only internal to whatever system we're working with. The angst over so-called "undefinable reals" goes away if you stop viewing the reals as a complete collection, pinned down in some platonic universe

>epsilon loops

buo

Why the fuck would the set not be {1, 2, 2, 3}, that's fucking retarded.

why would you count an element twice? thats fucking retarded

Because there's only one 2

I can understand what he's saying, but he could really do with phrasing it in model-theoretic formalism.

He bounces back and forth between a species being a collection of conditions given by a ZFC-formula and a species being its interpretation in a ZFC-model, which is a very poor abuse of notation.

In naive set theory, a principal consideration is not order, or accounting for multiplicity, etc. The first thing being asked is more fundamental: /is a given thing an element of a set, or not?/ Membership, versus non-membership. This is the first point.

Thus, union simply denotes "everything that's in either one of these sets (or, possibly, both)." Oh, 2 is in both sets while no other element among both sets has the same property, whoop-de-doo, that's not what we had just wished to investigate. What we have is that there are exactly three elements fitting the above criteria. And an extensive enumeration of "A U B" gives {3,1,2}, where the order of the list is in the first place irrelevant. We are concerned only with whether a given element belongs to the set or not, and this list suffices to express same.

In very-extra-naive set theory, if we for whatever reason were to encounter something like

{x,x,y,a,b,y,d,d}

Then it would always behoove us to eliminate whatever multiplicity has arisen from shoddy bookkeeping, to instead give something like

{x,y,a,b,d}. Indeed, this is essential if you want to know how many elements are in a set.

Johnny is in three private clubs: A sewing circle with 9 other members, the local jaycees with 34 other members, and a private BDSM group with 50 other members. Clearly no two of a given club's members are the same person. But of course, it may happen that a given person belongs to more than one club. In this case, it happens that these three clubs' membership are, as we say, disjoint with the lone exception of Johnny himself. to find the union of the three clubs, then, we would simply prepare an un-ordered list of (the) 94 names which counts Johnny exactly once.

Of course, some sets may be amenable to an ordering that we might like. and since Other Johnny is clearly not the same person as Johnny, we would have to account for distinct individuals having the same names, by designating them as distinct.

>ctrl+f "retard"
>13 results

Shit thread.

what else would you call op?

What? That's perfectly intuitive.

>this whole thread
Seriously.
I don't even have the words for the amount of retardation of some of these people, people who jerk off at math, but still can't grasp the idea that some system of axioms gives a certain result. Absolutely stupid. If it were a discussion about the axiom of choice, I wouldn't be saying anything, but really, if you cannot accept that {1, 2, 2, 3} and {1, 2, 3} are the same set because it doesn't fit your (very limited) common sense, then you should go to /b/ or something.
For fucking real people, if you're this much retarded, seriously, go

>uses \varnothing instead of \emptyset
thank jeebus

\emptyset is the worst.

Funny how the variations are both better and those commonly used. Another example is \varphi vs \phi.

It's because \varnothing is not defined in every font, and in some fonts \emptyset does actually look nice.

imo \phi is good for coordinates, \varphi is good for maps/functions