Russel's Paradox: Understanding Sets and Contradiction in Proofs

[Kniazi]

Prove Russel's paradox by contradiction and what does it tell us about sets?
I tried doing it like this and I am not sure it is right.

I supposed S was the collection of all sets and since S is a set S∈S.
Now we can split this universe S into two parts: U(for the unusual that are part of themselves like S) and N(for the normal sets).
So N={A∈S|A∉A}. In english this means that any set A will be part of N only if A is not an element of itself.
Now I plug in N in place of A, to apply the above statement for N.
N={N∈S|N∉N}.

So since N is not an element of itself, N belongs in N.

Is it correct ^?

Also what does this tell us about sets?

--
Khadija Niazi

 

[HallsofIvy]

Prove Russel's paradox by contradiction and what does it tell us about sets?
I tried doing it like this and I am not sure it is right.

I supposed S was the collection of all sets and since S is a set S∈S.
Now we can split this universe S into two parts: U(for the unusual that are part of themselves like S) and N(for the normal sets).
So N={A∈S|A∉A}. In english this means that any set A will be part of N only if A is not an element of itself.
Now I plug in N in place of A, to apply the above statement for N.
N={N∈S|N∉N}.

So since N is not an element of itself, N belongs in N.

Is it correct ^?

Also what does this tell us about sets?

What does "N belongs in N" tell us about N? What "paradox" does that give?

--
Khadija Niazi

 

[Kniazi]

Ok.
We assumed that N cannot contain sets belonging to themselves. After we plugged it into the statement, we got: "N is not an element of itself so N belongs in N". But that just contradicts the fact that N is not an element of itself? And that contradiction is the paradox.

 

[twoslit]

Prove Russel's paradox by contradiction and what does it tell us about sets?
I tried doing it like this and I am not sure it is right.

I supposed S was the collection of all sets and since S is a set S∈S.
Now we can split this universe S into two parts: U(for the unusual that are part of themselves like S) and N(for the normal sets).
So N={A∈S|A∉A}. In english this means that any set A will be part of N only if A is not an element of itself.
Now I plug in N in place of A, to apply the above statement for N.
N={N∈S|N∉N}.

So since N is not an element of itself, N belongs in N.

Is it correct ^?

Also what does this tell us about sets?

--
Khadija Niazi

You got the paradox right, although you should use language more carefully. "Part of" has no meaning in set theory. What you should say is, A will be an element of N just when A is not an element of itself.

What does this tell us about sets? That is an enormous question, with about a century of discussion buried in it. The short answer is, it doesn't tell us much about sets as such, so much as about how we can (and can't) define sets. It tells us that the naive idea, that we can simply define a set by saying "the set of all things that satisfy <some description of the things in the set>" , does not always work. Sometimes, what seems like a perfectly reasonable description just does not work as a way of defining a set.

So how can we know which descriptions are OK for describing sets and which aren't? There have been several attempts to answer this. The modern consensus is that sets are anything that is described by the axioms of set theory, usually Zermelo-Fraenkel set theory with the axiom of choice, commonly abbreviated as ZFC. The ZFC axioms basically say that various sets must exist if certain other ones do, eg the set of all subsets of a set; and that all sets must satisfy certain conditions, eg that one can't have infinitely 'deep' sets which have elements containing elements containing elements... and so on for ever. ZFC has been studied in intense detail and has been shown to be as consistent as any other set theory, and virtually all of mathematics has been formalized within it. But it still does not allow unrestricted 'descriptions' to define a set.

Sorry if this seems a bit vague, but this is a huge topic, and your question is very open-ended.

 

[Kniazi]

Twoslit, thank you so much, I really appreciate it. :)

 

[Stephen Tashi]

Also what does this tell us about sets?

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Russel's paradox tells you somehing about how particular sets can be defined. It shows that a certain approach to defining a set actually fails to define it.

In mathematics, one is supose to agree on some undefined terms and then define other statements using them. In ordinary conversation, we can philosophize about how a term ought to be defined. Philosophizing involves speaking about the term as if it already means something and then trying to figure out how to state that meaning. If we consider the question "What is a set?" then mathematically we aren't asking a specific question unless "set" is is already defined - and if "set" is already defined, then the question is answered.. Philsophically, we may ask "What is set?" and try to figure out how to precisely define our intuitive idea of "set" in mathematical terms.

One mathematical approach is to take ""set" and "element" as undefined terms and "x is an element of S" as an undefined relation. To define a particular set T, it might seem adequate to give a rule that can be used to determine whether any element is in T. For example, the typical "set builder notation" gives a statement and all elements that make the statement true are in the set being built. Russels paradox shows an example where the "set builder" method doesn't build a set.