Re: Galileo's Paradox and the Project of the Reals

In article <45a2eb91@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

stephen@xxxxxxxxxx wrote:
David Marcus <DavidMarcus@xxxxxxxxxxxxxx> wrote:
Mike Kelly wrote:
Tony Orlow wrote:

such that it is referred to as set size.
Which theorem or definition of set theory refers to "set size"?

You STILL seem to be confused between how people describe set theory
and what it actually says.

Set theory says there are sets that can be bijected with some of their
subsets. Do you accept that?
Sure.
So you're just arguing about terminology. You accept what set theory
actually *says*, which is that some sets can be bijected with their
subsets. And, for example, that the evens can be bijected with the odds
and both can be bijected with the naturals.

Using set theory doesn't require calling cardinality "size". People
call cardinality "size" becase it makes intuitive sense to most people
to think of it as size. I must've told you a dozen times that if you
wish you can just replace "has equal cardinality to" with "is
bijectible with" and nothing is changed. You don't have to "accept
proper subsets being the same size as their supersets" because set
theory doesn't say that.

I'd like an acknowledgement from you that set theory doesn't say
anything about "size". You've been told this lots of times but keep
repeating the same garbage.

A reasonable request.

But one which has been ignored for almost two years now. I doubt
this year will be any different.

Stephen

Am I incorrect in saying there are half as many evens as naturals, or
the square root as many squares as naturals, or log2 as many powers of 2
as naturals? If you claim those statements contradict your theory, then
you are de facto claiming that transfinite cardinality IS set size.

"As many as" is the commonplace for equinumerous, wish is in turn, the
same as admitting a bijection.

This is trivially true for finite sets, and by obvious extension, for
all sets.
.

Relevant Pages

  • Re: Anti-diagonalist page
    ... that axiomatic set theory just ignores I wouldn't find ... the number of naturals, except when they're speaking ... very informally - instead they say that the cardinality ... For Dedekind defined infinite sets as those that could be ...
    (sci.logic)
  • Re: An uncountable countable set
    ... I'm not sure how this would be proven in set theory (I don't think it ... In ZF it is NOT the case that every set must have a cardinality ... required to list the naturals in binary defies classification in this ... infinitely rare for infinite n, so the result that TO object to is quite ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... That does not make it rely on anything more than set membership. ... Using set theory doesn't require calling cardinality "size". ... When it defines "equinumerous" as meaning having a bijection between ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... So construct a bijection ... and the elements of the finite field of order n. ... Collapse of Toy Set Theory. ... say now that finite ordinals are _identical_ with naturals, ...
    (sci.math)
  • Re: An uncountable countable set
    ... I'm not sure how this would be proven in set theory, but it appears to be a belief, anyway, that all sets can be classified through cardinality. ... so there must be a cardinality for the set of bit positions in the naturals. ... An extra bit doubles the number of strings, so if you have gotten to the least bit string necessary for n naturals, you will have at most n-1 unused strings. ...
    (sci.math)