Re: Is zero even or odd?
From: Fred Bloggs (nospam_at_nospam.com)
Date: 12/24/04
- Next message: Fred Bloggs: "Re: Is zero even or odd?"
- Previous message: Torkel Franzen: "Re: Is zero even or odd?"
- In reply to: Dave Seaman: "Re: Is zero even or odd?"
- Next in thread: Dave Seaman: "Re: Is zero even or odd?"
- Reply: Dave Seaman: "Re: Is zero even or odd?"
- Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
Date: Fri, 24 Dec 2004 21:36:20 GMT
Dave Seaman wrote:
> On Fri, 24 Dec 2004 15:22:55 GMT, Fred Bloggs wrote:
>
>
>
>>Torkel Franzen wrote:
>>
>>>Fred Bloggs <nospam@nospam.com> writes:
>>>
>>>
>>>
>>>>... because the non-existence of infinity strictly between countability
>>>>and first uncountability ( power set of countability) has been shown to
>>>>be equivalent to the Axiom of Choice.
>>>
>>>
>>> You're mistaken about this. Why these ill-informed exchanges in all
>>>these unrelated groups?
>>>
>>
>
>
>>Are you saying this has not been established yet?
>
>
> You made so many mistakes in that one sentence that it's hard to know
> where to begin.
>
> For one thing, the nonexistence of cardinals strictly between aleph_0 and
> aleph_1 is a matter of definition. Without the axiom of choice, the
> possibility exists that there may be cardinals that are not comparable
> with either of those, but there still can't be any that are strictly
> between. In other words, aleph_1 is certainly minimal among the
> uncountable cardinals, even without AC.
As you said aleph_1 is minimal by definition, and without AC it may not
be a bound of all the uncountable infinities- there may be an infinity
of aleph_1's- the ordering is not total. Interesting that you say AC->
the ordering of the cardinals will be total.
>
> For another thing, the power set of the naturals has the same cardinality
> as the reals, namely 2^aleph_0 = c, the cardinality of the continuum.
> The assertion that c = aleph_1 is called the Continuum Hypothesis (CH).
I think Cantor's original statement was that there is no infinity
strictly intermediate to countability and c- which of course means
c=alepha_1 if it exists.
> Not only has it not been established "yet"; it's been established that CH
> will never be proved or disproved in ZFC (Zermelo-Frankel set theory plus
> the axiom of choice).
Right- it is undecidable- that much has been proved in ZFC- the CH is
independent of AC. Supposedly H. Woodin has constructed a plausible
axiom recently which if incorporated into ZFC implies the CH is false.
Nonetheless, it makes no sense to speak of anything as being "true" as
either system is self-consistent, the assumption of CH or /CH will never
lead to a contradiction of the axioms.
- Next message: Fred Bloggs: "Re: Is zero even or odd?"
- Previous message: Torkel Franzen: "Re: Is zero even or odd?"
- In reply to: Dave Seaman: "Re: Is zero even or odd?"
- Next in thread: Dave Seaman: "Re: Is zero even or odd?"
- Reply: Dave Seaman: "Re: Is zero even or odd?"
- Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
Relevant Pages
|
