Russel's Paradox in Naive Set Theory

[EricJRose83]

I realize that Russell's Paradox in naive set theory is considered to be, well... a paradoxical fallacy. Despite the fact that it is paradoxical and goes against logical intuition, is it really illogical though? It seems to me that the method in which the paradox arises is perfectly sound and as a result, the paradox should be taken as an inherent aspect of logic, instead of being shunned and 'renormalized' as it was in axiomatic ZF set theory.

 

[HallsofIvy]

The whole point of a "paradox" is that you can show that a given statement and its negation are true. Once that is true, it follows that you can "prove" any statement at all.

What good is a logic in which every statement can be proved true?

 

[EricJRose83]

I don't think you can really 'prove' any statement considering the axiomatic foundations used to justify those statements can't be proven themselves. Also, I realize that viewing paradoxes as an inherent real property isn't very useful considering everything, but I don't think that should take away from the fact that they might possibly offer revelations on the true nature of certain logical systems. I hold truth to be of greater virtue than usefulness. Usefulness aside, do you think Z & F were really justified in giving set theory it's axiomatic base to do away with these paradoxes?

 

[SteveL27]

What good is a logic in which every statement can be proved true?

It's complete :-)

 

[EricJRose83]

It's complete :-)

This :)

 

[bobby2k]

Hi

I must admit I have not read much about this, but I can give my two cents. From what I have understand when you make a theory with axioms, you must allways be sure that the next axioms and definitions does not contradict the earlier ones. Also logic is just a tool for what your theory, I do not think you can say it is a part of the theory.
So set-theory is based on a few axioms, and the logic is the tool that are used to build set-theory. You can try to say that there can be a set that contains all other sets, because you allready have an axiom called the "axiom of specification", which allows you to make the subsets used in Russels paradox. But this implies the contradiction.

Also take this with a grain of salt. But I think that Russels paradox shows the contradiction with the axiom of specification. Because if you can make a subset of A where the elements of P(x) is true, where x are elements of A, you can also make a subset of B where ~P(x) is true. And using the rules of logic every elements in the main set must be in one of these subsets. But the set in Russels paradox is in none, hence it contradicts the axiom of specification.

from wikipedia:
"An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms."
"An axiomatic system will be called complete if for every statement, either itself or its negation is derivable."
http://en.wikipedia.org/wiki/Axiomatic_system#Properties

Maybe it is complete as you say, but not consistent.

 

[DrewD]

...I realize that viewing paradoxes as an inherent real property isn't very useful considering everything, but I don't think that should take away from the fact that they might possibly offer revelations on the true nature of certain logical systems.

The revelation is that naive set theory is inconsistent. I don't think there is much more value from that. Every statement is true and every statement is false. Granted, this would make Analysis proofs much easier...

 

[aleph-aleph]

Naive set theory is defined using daily language because the mathematics at that time has not been formalised yet. Naive set theory has a lot of ambiguity because of the impreciseness of language. Perhaps some misinterpretation occur when mathematicians axiomatised set theory.

 

[SteveL27]

Naive set theory is defined using daily language because the mathematics at that time has not been formalised yet. Naive set theory has a lot of ambiguity because of the impreciseness of language. Perhaps some misinterpretation occur when mathematicians axiomatised set theory.

But there is a bit of a double meaning in the phrase. Halmos's classic text Naive Set Theory is very commonly used in the undergrad math major class on set theory. There's nothing vague or contradictory in that book.

So at least one author of a prominent textbook thinks there is value in using the term.

The other standard text is Suppes's Axiomatic Set Theory. I never had the chance to look at that one. Is it rigorous in ways Naive Set Theory isn't? I'm sure Halmos starts with the proper rules for set formation and commits no paradoxes.

 

[aleph-aleph]

I agree that there is nothing contradictory in Halmos (1960), just my random thought that there might be a misinterpretation since Cantor himself didn't bothered by the paradoxes and his original papers are difficult to read and understand.

Check these out,
http://www.santarosa.edu/~jomartin/IrratFiles/Cantors1874Paper.pdf
http://www.scribd.com/doc/57083858/Cantor-Grundlagen

I think it's entirely possible to misinterpret Cantor's work while the interpretation itself is logically consistent.