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/sci/ - Science & Math

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7130979 No.7130979 [Reply] [Original]

What is the set of all sets?

>> No.7130981

muh dik

>> No.7130984

the set of all crime committed by blacks is uncountably infinite

>> No.7130985
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7130985

>> No.7130990

Isn't it the hollow set? (Dunno how it's said in english, I did a literal translation)

>> No.7130991

>>7130979
For any universe U there exists a universe U' containing U and all objects in U.

>> No.7130993

>>7130991
prove it

>> No.7130998

>>7130993
I don't need to.

>> No.7131003

>>7130993
It's an axiom, dip.

>> No.7131005

>>7130979
There can not be a set of all sets. The set of all sets would need to contain itself and therefore cannot exist. This principle is known and the incompleteness theorem.

>> No.7131006

>>7130979
Isn't that kind of construction forbidden in set theory?

>> No.7131014

>>7131005
>the principle is known and the incompleteness theorem
The two are unrelated. The incompleteness theorems (plural) concern models of theories. You just recited Russel's paradox.

>captcha: Helpo; because my Mexican savior must save me from this thread

>> No.7131030

>>7130993
Is an axiom consistent with set theory*


*assuming a certain class of large cardinals.

>> No.7131037

>>7130991
Why is "and all objects in U" necessary

>> No.7131234
File: 1.05 MB, 400x296, 1403514332578.gif [View same] [iqdb] [saucenao] [google]
7131234

>>7130985

>> No.7131662

>>7131037
Because U cannot contain itself, so we are axiomatizing the existence of a universe that contains U as well as everything in it. It's necessary so that consistent choice of U' will always give us a well-ordered universe of universes.

>> No.7131679

>>7131005
>The set of all sets would need to contain itself and therefore cannot exist.
Don't all sets contain themselves?

>> No.7131687

>>7131679
Yes. That guy is just buzzwording and didn't prove anything.

>> No.7131696

>>7130979
the only set that contains itself
>mind=blown

>> No.7131713

>>7131030
The problem with the question is in assuming that class of large cardinals; we aren't in NBG here, where we would just say that all sets are inside of a proper class. I'm using Grothendiek's approach. He already showed the consistency of his theory.

>> No.7131723

>>7131696
What about the set that only contains itself?

>> No.7131744

>>7130979
Upset

>> No.7131861

>>7131234
(╥﹏╥)

>> No.7131960

>>7131713
The universe axiom is equivalent to the existence of a cofinal class of strongly inaccessible cardinals ( this means that for every cardinal, there is a strongly inaccessible cardinal bigger than you ordinal ). Gothendrieck theory is consistent because of this fact.

>> No.7132127

>>7131960
Oh, I didn't realize that was why it was consistent. Thanks!

>> No.7132271

What is the cardinality of U ?
What is the cardinality of P(U) ?

>> No.7132280

>>7131005
100% wrong. Sets include themselves by definition.

>> No.7132288

>>7132280
So what you are saying is <span class="math">\forall S \left ( S \epsilon S \right )[/spoiler]
Or am I misunderstanding something?

>> No.7132298

>>7132288
<span class="math">S \subset S[/spoiler] not <span class="math">S \in S[/spoiler]

>> No.7132304

>>7130979
arc-O
integral of O

>> No.7132305
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7132305

>>7132298
>S is a strict subset of S

>> No.7132313

>>7132305
<span class="math">S \subset S[/spoiler] only means strict in shitty countries.

>> No.7132316

>>7132313
Every textbook is a shitty country.

>> No.7132338

>>7132305
>>7132316
It's standard notation for inclusion, not strict inclusion.

>> No.7132343

>>7132280

100% wrong. Sets are subsets of themselves by definition but do not contain themselves.

>> No.7132348

>>7132338
I have literally never seen that used to mean inclusion. I have only seen that used to mean strict inclusion, or implication in some symbolic logic textbooks.

>> No.7132350

That's a big set.

>> No.7132355

>>7132348
Well I'm pretty sure I've never seen anyone use <span class="math">A \subset B[/spoiler] in that case, they'd rather use <span class="math">A \subsetneq B[/spoiler].

>> No.7132359

>>7132348
I'm sorry you've been exposed to notation which is not standard and are doomed to make a fool out of yourself contradicting a notation that is the norm for the rest of your life.

>> No.7132361

>>7132355
Forgot 4chan doesn't have all symbols. Granted I live in Europe so it might be different around the world but I just grabbed some random books around me and both Lang and Lee use <span class="math">A \subset B[/spoiler] for general inclusion.

>> No.7132374

<span class="math">subset[/spoiler] is the standard for general inclusion, but I also think it would be better, if more elaborate, to use \subseteq.
I personally use <span class="math">subset[/spoiler] for strict inclusion in my notes.

>> No.7132379

>>7132350
for a Grothendieck universe U

>> No.7132385

>>7132359
1/10

I personally prefer to use the subset symbol as strict, instead of subsetneq, and use subseteq for general subset.
However you're using general subset most of the time, and often one will write A subset B, A not equal B instead of A subsetneq B so I'm thinking of switching my notation just for speed. It's usually clear when it can't be equal anyways.

>> No.7133324

>>7132350
For <span class="math">\mathbb{U}[/spoiler]

>> No.7134285

>>7131234
spiders don't usually creep me out but shit man

>> No.7134286

>>7133324
Hey I made that joke 3 lines above you.

>> No.7134297

>>7134286
<span class="math">\overline{\overline{\overline{\mathbb{U}}}}[/spoiler]

>> No.7134307

>>7134297
<span class="math">\overline{\overline{\overline{\math bb{U}}}}[/spoiler]

>> No.7134310

>>7133324
<span class="math">\overline{\overline{\overline{\mathbb{U}}}}[/spoiler]

>> No.7134316

>>7134307
Unknown control sequence '\math'

>> No.7134317

>>7134297
>>7134307
>>7134310
put spaces in your code or 4chan will do it for you

<span class="math"> \overline{ \overline{ \overline{ \mathbb{U}}}}
[/spoiler]

>> No.7134321

>>7134317
Thank you!

>> No.7134338

>>7134297
whoaaa
how'd you do that

>> No.7134343

>>7134338
My failure to LaTeX ruined the joke, but three lines above you (U)... har har...

>> No.7134449

>>7132280
Consider the set of integers.

Is the set of integers an integer?
Of course not.
So "the set of integers" is not contained within "the set of integers".

I think you meant "sets ARE SUBSETS of themselves by definition", but basically you have a highschool level understanding of set theory, and a poor one at that.

>> No.7134465

>>7132348
>>7132316
>>7132305


It's a cultural thing, and an archaic convention in countries that generally use it for strict subsets.

i.e. if you've ever read a book published before the 90's, which you probably haven't, as you're clearly an undergrad, you'll have seen it all over.

I could quote you several of the old editions on my bookshelf that use this notation of a non-strict subset if you want proof.

>> No.7134473

>>7134465
Wasn't trying to be confrontational or anything. However, this does give us a very clear example of the utility and convenience of having universal conventions with regards to notation.

>> No.7134476

>>7134473
Yep. but unfortunately that never happens anywhere in maths.

>> No.7134492

>>7134476
And to boot, old conventions are constantly undergoing change. That's what you get when you let short-lived organisms run your formal science, though.

>> No.7134496

>>7131679
>Don't all sets contain themselves?
no. of course not.
e.g. A = {1,2,3} is not equal to A = {A,1,2,3}

even though it makes no sense one could just go with it as people did with imaginary numbers

>> No.7134498

>>7132348
>I have literally never seen that used to mean inclusion. I have only seen that used to mean strict inclusion, or implication in some symbolic logic textbooks.
im from germany and have seen some authors mentioning it can mean either
confuses me every time i see it tbh

>> No.7134527

>>7134496
>makes no sense
You never took programming, did you ?

>> No.7134588

>>7131005
You're thinking of the set of all sets that do not contain themselves.

A set is allowed to contain itself. Say you have a set of sets of integers, the contents of the set are integers so the set itself is also follows the rule of membership.

>> No.7134595

>>7134588
I don't get it.
the contents of the set are sets (of integers).

>> No.7134611

>>7134595
On second thoughts I think a better example would be the set of integers and sets containing integers. The set itself would then contain integers and therefore be a set containing integers and therefore be a member of itself. The point is that sets can be members of themselves.

What was almost explained was Russell's Paradox, which is: "If R is the set of all sets that do not contain themselves then is R a member of itself?" Each of the two possible answers then implies the opposite answer. The set of all sets as the original post said would contain itself by definition because it is a set so it is part of "all sets".

>> No.7134619

>>7134588
>>7134611
A set can only contain itself through infinite recursion.

I'm not a mathematician or anything, but someone once told me that sets are forbidden from containing themselves in modern set theory

>> No.7134631

>>7134619
Of course a set can contain itself. Take a set where the rule to be a member is to be blue or to be a set that contains blue things. All blue things will be a member of this set, meaning the set will then contain blue things, meaning that it has the same property that all the other members have in common, making it a member of the set itself.

A set is a collection of things that follow a rule, it is possible for a set to be consistent with its own rule.

http://en.wikipedia.org/wiki/Russell%27s_paradox

>> No.7134639

>>7134631
I didn't say it was impossible to construct in that way.
I just said it requires infinite recursion, just as your example does.

But to be honest I don't know why I brought that up. My main point was
>I'm not a mathematician or anything, but someone once told me that sets are forbidden from containing themselves in modern set theory

If it's forbidden in modern set theory then that's that, you can't do it because it's not allowed. But I qualified it with "I'm not a mathematician" because I'm not 100% sure if it is or not, just repeating what I was told

>> No.7134642

>>7130979
The power set

>> No.7134651

>>7134639
I don't think that's infinite recursion, if anything it's singular recursion. You check that the rule applies to the "initial" members of the set then you move up one level and see if it applies to the set, you conclude that the set is therefore a member of itself and hey, you're done, you've covered everything that can be a member of that set. There's nothing infinite about that.

My point is that the statement: "There cannot be a set of all sets because the set cannot contain itself" is wrong because a set can clearly contain itself. I think the poster is simply misquoting Russell's Paradox which is all about sets containing themselves.

The paradox shows that standard set theory, from which Russell and Whitehead effectively derived mathematics in Principia Mathematica, contains an inconsistency despite being a complete set of axioms. The Incompleteness Theorem states that there cannot be a both consistent and complete set of axioms. That is, in a complete set there will always be a paradox and/or an unprovable axiom.

>> No.7134652

>>7134639
You're thinking about the axiom of foundation, which indeed forbids that kind of things. Some people also sometimes use its negation (anti-foundation), or just not use it

>> No.7134658

>>7134651
But the actual construction of the set must be infinitely recursive
Because if A contains A, then that A also contains A, which also contains A, etc..
Basically it's impossible to explicitly write out all the elements (and elements of their elements) of any set that contains itself

But again I wasn't really trying to make any point with that, I just brought it up for the hell of it, it doesn't mean that such a set can't exist or anything

>> No.7134667

>>7134658
>Because if A contains A, then that A also contains A, which also contains A, etc..
When you decide that a set A follows its own rules then you don't have to create a new A to contain the old A. It's not like you have the contents of the set and, draw a line around them, then have to draw a line around the first line, then another line around that and so on. The container is not entirely independent from the contents as it were.

>Basically it's impossible to explicitly write out all the elements (and elements of their elements) of any set that contains itself
It's impossible to explicitly write out all the elements of the set of all positive integers.

N = {1, 2, 3, 4, 5, 6, .......... }

It's still allowed to be a set though, so why can't a set containing itself be a set?

Either way, the statement made earlier ITT is a misquote of something a bit different to what he thought he was talking about.

Have a read about Russell's Paradox and Goedel's Incompleteness Theorem if you want to learn about this properly. A /sci/ thread isn't really the best place.

>> No.7134676

>>7134651

>Incompleteness theorem

Of course you can have a consistent, complete theory extending PA, however any set of axioms of that theory can't be algorithmically definable.

>> No.7134686

>>7134527
>You never took programming, did you ?
i did, but you obviously never experienced an infinite recursion

>> No.7134704
File: 60 KB, 761x306, Bildschirmfoto 2015-03-17 um 01.03.55.png [View same] [iqdb] [saucenao] [google]
7134704

>>7134631
The axiom that effectively forbids sets to contain themselves is Regularity
en.wikipedia.org/wiki/Axiom_of_regularity

For the "obscure" theories which allow for it (as the guy above me mentioned), see Non-well-founded set theory.
en.wikipedia.org/wiki/Non-well-founded_set_theory

>>7134651
>despite being a complete set of axioms
Russels paradox is something much simpler than Gödels incompletness theorem, and you're using completeness strangely. The syntactic completeness the theorem is about is about provability.

>>7134527
In computer science,
-you may either construct an object A, in which case you can't put A in itself,
-or you put a reference a in it, for which you have a function f with f(a)=A (but then A isn't really in A, just a name that refers to A)
-or you have a language with comprehension (as you seem to be suggesting).
http://en.wikipedia.org/wiki/List_comprehension

I only know list comprehension for statically typed language, and there at least, a list of lists of infinite death isn't syntactically allowed.
That being said, implementing sets effectively in a programming language, an object for which {a,b,c,d,b,a}={a,b,c,d}, is hard. Harder still, is to implement sets which aren't lists.
For example, you generally deal with a some choice, i.e. the terms of a set when implemented in a computer are usually a prior orderable :/

>> No.7134725

>>7134704
btw. the code in the pic is a veiled version of the Haskell code

x = 1 : map (2*) x

which is possible due to the languages lazy evaluation feature.
It says
>let x be the list which starts out with 1 and is followed by a copy of the list, where each entry is multiplied by 2.
When you ask Haskell to return the fourth entry of x, it will deduce that it's 2 times the third entry of the list, which which is two times the second entry of the list, which is two times the first entry of the list, which is 1. So it returns 8.

>> No.7134728

>>7134667
>It's still allowed to be a set though, so why can't a set containing itself be a set?
Again, I never said it wasn't. I've said that three times now.

>When you decide that a set A follows its own rules then you don't have to create a new A to contain the old A
that's irrelevant. If the set contains itself then it's elements are infinitely recursive within that element of itself, regardless of how it was constructed.
If it's not, then the sat cannot be said to contain itself.

>> No.7134733 [DELETED] 

>>7134667
>>7134728
In other words, all i was saying was this
If A = {A}
then A = {{A}} = {{{A}}} = {{{{A}}}} = {{{{{A}}}}} = {{{{{{A}}}}}} = etc...

as just the most simple example.
It's infinitely recursive. All sets that contain themselves are like that. I didn't say this means the set can't exist or anything, I was just pointing it out

>> No.7134871

>>7134728
>that's irrelevant. If the set contains itself then it's elements are infinitely recursive within that element of itself, regardless of how it was constructed.
question is: so what? it is purely abstract so why is this a problem?

>> No.7134897

>>7134686
neither did you, technically

>> No.7134922

>>7134897
>neither did you, technically
i saw my program crash when i tried one

>> No.7134937

Let S be the set which countains all the other sets.
Let S2 = {S}

card(S2) > card(S) => S doesn't countain all the other sets

>> No.7134940

>>7134937
#rekt

>> No.7134944

>>7134937
the cardinality of S2 is 1.

>> No.7134952

>>7134937
Shouldn't S2 = P(S)?

>> No.7134953

>>7134944
Nope, because S is just a convenient symbol. You can replace it with what it is, aka every single set possible, but that would take quite some time

>> No.7134959

>>7134937
You can replace the cardinal thing with S is in S2 but S2 isn't in S QOD

>> No.7134962

>>7134953
>>7134937
Your proof is wrong.

>>7134952
This is why.

Also, {S} is a set which contains only 1 element, S. So it's cardinality is 1.

>> No.7134975

>>7130979
wouldn't the set of all sets be the universal set? but the universal set doesnt exist in ZF due to the regularity axiom

>> No.7134981

whats a set

>> No.7134985

>>7134975
The existense of an universal set is contradicted by foundation axiom, however the purpose of this axiom was not to aviod Russel's paradox, but to give a 'nice structure' to the universe. The existense of an universal set is contradicted by a weak instance of the comprehension axiom.

>> No.7135279

>>7130979
>the set of all sets
... is an ill-defined, vague concept, therefore not a set.

>> No.7135683

>>7130979
A Klein bottle.
Don't ask how.

>> No.7137082

>>7130984
Until you compare it to the set of all crime committed by whites.

>> No.7137086
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7137086

>>7137082
0/10 learn to math better otherwise you won't be funny.

>> No.7137114

>>7130979
Universe

>> No.7137159

>>7134492
>>7134476
Do you know what the best thing about mathematical notation is? It's not economics notation.