Russell's Paradox

[Anonymous No.9979297]

How many of you never bothered do check this?

Let [math]X=\{Y:Y\not \in Y\}[/math].
[The set of all sets not members of themselves]
1. if [math]X \in X[/math] then [math]X \not \in X[/math] is false and [math]X \not \in \{Y:Y\not \in Y\}=X[/math].
2. if [math]X \not \in X[/math] then [math]X \in \{Y:Y\not \in Y\}=X[/math].
3. [math]X \in X[/math] iff [math]X \not \in X[/math] follows.

The statement [math]X \in X[/math] then is both true and false,
which makes naive set thoery unsuitable without proper precautions.

More on this here:
https://plato.stanford.edu/entries/russell-paradox/#ERP
https://en.wikipedia.org/wiki/Russell%27s_paradox

[Anonymous No.9979390]

This shit was like day 2 of my discrete math class.
Just need to patch naive set theory to prohibit circular definition of sets.
Don't know why it causes undergrad philosophers so much trouble.
>here is a theory
>"i found a case for which the theory does not have a clear answer"
>thanks bert, nobody would ever really need an answer for that case anyway, we will modify the definition to prevent it

[Anonymous No.9979395]

>>9979390
>class
you realize they invented this concept specifically to avoid russell's paradox

anyhow, you brainlets should read up on Gödel.

he took these concepts, plus other concepts from Russell, and made several theorems which in principle could be used for major breakthroughs in mathematics

[Anonymous No.9979420]

>>9979297
>Russell's paradox
You are like a little baby

[Anonymous No.9979434]

>>9979297
>If it's math, it can't have mistakes.
proofs?

[Anonymous No.9979449]

Its one of the most popular paradoxes in math. Wtf are you talking about?

[Anonymous No.9979465]

>>9979395
>you realize they invented this concept specifically to avoid russell's paradox
Can you help me figure out how classes neutralize the Russel's Paradox?
I have elaborated in pic related.
Now, isn't 1. a contradiction?

[Anonymous No.9979468]

>>9979465
honestly, no, my math training stopped before we really analyzed how classes work

all i learned is that "the set of all sets which do not contain themselves as a member" is somehow a class, not a set. and therefore classes are better than sets for some mathfag reason

i leave it to mathfags to explain this very-characteristic shifting of the goalposts

[Anonymous No.9979499]

>>9979468
Isn't it obvious? "the set of all sets which do not contain themselves as a member" cannot exist since if it contains itself then it cannot contain itself, and if it does not contain itself then it must contain itself. A class solves this

[Anonymous No.9979509]

>>9979499
sure, i see the contradiction, and i do see how if you shift the goalposts to say "oh wait my set definition has some contradiction, so i introduce a new concept called sets" could fix this, but i don't know the details of how your new thing "class" works that makes it immune from the contradictions inherent in sets

honestly, my impression is that it looks like a chicken-and-egg problem, and you just go one generation up, but fine, i trust mathematics and i trust you guys can explain it to me

[Anonymous No.9979512]

>>9979509
A class cannot contain other classes

[Anonymous No.9979514]

>>9979297
>>9979499
sure, i see the contradiction, and i do see how if you shift the goalposts to say "oh wait my set definition has some contradiction, so i introduce a new concept called classes" could fix this, but i don't know the details of how your new thing "class" works that makes it immune from the contradictions inherent in sets

honestly, my impression is that it looks like a chicken-and-egg problem, and you just go one generation up, but fine, i trust mathematics and i trust you guys can explain it to me

[Anonymous No.9979515]

>>9979509
A class cannot contain a class

[Anonymous No.9979517]

>>9979514
A class cannot contain a class

[Anonymous No.9979518]

>>9979512
for the record, the deleted post this guy is quoting is because i deleted and reposted this post
>>9979514
with a typo fix

[Anonymous No.9979529]

>>9979517
okay, a class cannot contain a class. fine. that resolves russell's paradox

the lesson from russell's paradox, however, is that definitions of things, even very old things, might actually not be well-defined

you learn more, and you see the contradiction in your definitions

this doesn't mean mathematics is bunk, it only means that work needs to be done even on the fundamentals.

you advance the peak, most applicable-to-science-and-computers-and-sociology stuff, and often when people poke holes in your stuff, it means there is some hole even deeper down in the fundamentals. so there needs to be research at all levels. otherwise we're relying on an ancient, rickety old artifice of old and busted ideas

[Anonymous No.9979534]

>>9979529
ZFC resolves Russell's paradox, so there are no holes in the fundamentals

[Anonymous No.9979545]

>>9979534
take a step back and you might realize that the point i’m making is at the “meta” level. i’m sure there will be future developments on the fundamentals of math, and insisting they are absolutely correct already misses the lesson that russell’s paradox provides

[Anonymous No.9979567]

>>9979545
Then your point just boils down to godels incompleteness theorem, so I don't know what's your point. Do you have anything to add or did you just want to talk about it?

[Anonymous No.9979590]

>>9979517
>A class cannot contain a class
You don't need the regularity axiom for showing that no contradiction arises from the Russell's Paradox.
The class building axioms are sufficient.

[Anonymous No.9979598]

>>9979449
OP is literally 12 though.

[Anonymous No.9979599]

>>9979590
A class not being able to contain a class doesn't come from the axiom of regularity. Where did I say otherwise?

[Anonymous No.9979611]

Alright you fucking brainlets listen up because I will explain it to you.

First there is all the things which can be defined. Great. The most general way to classify them is in a collection. A collection can contain itself or not; it doesn't matter. A collection is just a bunch of things which can be defined. It can contain itself. The resulting "paradoxical" collection is still a collection even if it is not something which exists in the universe.

Now, a class is simply a collection of all things which have a specific characteristic, that which defines the class. A class can contain itself. A class can contain itself. A proper class cannot.

Finally a set is a class of unique elements. Basically a bunch of elements which are unique in the set.

[Anonymous No.9979613]

>>9979599
Axioms are relative to indicidual theories. All objects are simply models within theories built on axioms. We all know what the axioms of ZFC set theory are. This conversation is meaningless without a set of axioms (and a system supported by those axioms) defining class theory. The notion of "universal axioms" is retarded.

[Anonymous No.9979619]

>>9979613
Where did I bring up the notion of universal axioms? How is this post related to my post?

[Anonymous No.9979620]

>>9979599
>Where did I say otherwise?
You prove that using the regularity axiom.

[Anonymous No.9979626]

>>9979619
You didn't. It's not.

[Anonymous No.9979638]

>>9979567
close but no, the incompleteness theorem says that there are true but unprovable statements in any mathematical system that includes arithmetic

OTOH what i'm saying is that history teaches us that the definitions we've used to build up mathematics might not really be well-defined -- maybe what we have right now is plain wrong due to some overlooked counterexample

if you followed the analogy from russell's paradox to what i was saying, you would see how it's analogous. but whatever, you can't expect mathtards to use their brains for anything besides 100% linear thinking

[Anonymous No.9979662]

>>9979599
>A class not being able to contain a class doesn't come from the axiom of regularity.
If [math]A \in A[/math] then [math]A=A[/math] and [math]A \in A[/math],
or [math]\exists X[/math] (which is [math]A[/math]) s.t. [math]X=A[/math] and [math]X \in A[/math].
Hence [math]\exists X[/math] s.t. [math]X \in A\cap \{A\}[/math], which is [math]\emptyset[/math] by the axiom of regularity.

[Anonymous No.9979745]

Holy smokes this thread is full of brainlets and half-correct answers.

>>9979390
>answer
That's not how axiomatic theories work.

>>9979465
>>9979468
There are a few different axiomatic set theories that define classes differently and not all set theories do it (eg. ZFC).
NBG Set Theory has probably the simplest notion of a class. Basically all sets are classes but not all classes are sets. Therefore when you define something like
>Let S be the class of all sets
it's done so in such a way that S itself is not a set and therefore is not contained in S. For details refer to:
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory

>>9979529
>the lesson from Russell's paradox, however, is that definitions of things, even very old things, might actually not be well-defined
Russel's paradox was discovered not long after the theory had been proposed but it was after a lot of work had gone into developing the theory so it was very much a surprise to many. In general though you're right that it's a warning about possible inconsistent theories.

>>9979499
>Teaching like a brainlet
The problem isn't the people you're explaining things to, it's you.

>>9979534
>ZFC resolves Russell's paradox
It does.
>so there are no holes in the fundamentals
This statement is not implied by the previous one.

>>9979567
No, stop talking about things you know little about.

>>9979598
>>9979613
>>9979638
This.

>>9979611
>Hurling foundational definitions without an axiomatic system.
wew lad

>>9979662
>/pol/esmoker picture in /sci/