Re: Why is the Russell Paradox necessary?

On 5 Mar 2007 19:31:57 -0800, "Calvin" <crice5@xxxxxxxxxxxxxx> wrote:

In ordinary set theory, in which Russell's Paradox is used
to show that assuming the existence of a set of all sets
leads to that paradox, ie. a contradiction, I wonder why
the Russell Paradox method is necessary.

This isn't what Russell's paradox shows.


We know that the cardinality of the set of all subsets of a
given set is greater than the cardinality of the given set.
<snip>

The proof of this employs a method which is essentially the same as
Russell's paradox. Plus you have to define cardinality, which means
you need to know about bijective functions, and use more tools than
are necessary.

Russel's paradox, on the other hand, is simple, and uses nothing than
the definitions to show why a set theory with unrestricted
comprehension is inconsistent.

But again, you're talking about two different results...

Thanks,
Mark
.

Relevant Pages

  • Re: ZFC is proven to be inconsistent
    ... I asked you to prove the Skolem paradox is a contradiction ... the very axioms which prove the existence of uncountable sets be satisfie ... cardinality of the real numbers exceeds the cardinality of the naturals ...
    (sci.logic)
  • Re: ZFC is proven to be inconsistent
    ... I asked you to prove the Skolem paradox is a contradiction ... the very axioms which prove the existence of uncountable sets be satisfied ... cardinality of the real numbers exceeds the cardinality of the naturals. ...
    (sci.logic)
  • Re: Why is the Russell Paradox necessary?
    ... leads to that paradox, ie. a contradiction, I wonder why ... the Russell Paradox method is necessary. ... given set is greater than the cardinality of the given set. ...
    (sci.math)
  • Re: Indefinite Extensibility and Computationalism
    ... Your argument is too fuzzy to take seriously without clarification. ... It's not enough to agree that "heterological" leads to a contradiction, ... That leads to a contradiction. ... A paradox can be provoked by the assumption that a concept ...
    (sci.logic)
  • Re: Mathematicians are in deep shit for 2 reasons
    ... Because in idiomatic English a person saying ... "Skolem's Paradox sounds like a contradiction" ... You mean, "it sounds like a contradiction, although it's not one", ... So, while Skolem's Paradox sounds like an antinomy, it is in fact ...
    (sci.logic)