Re: Distinct linear orderings on Z

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/24/05 

Date: Thu, 24 Mar 2005 11:13:19 -0500

stephen@nomail.com said:
> In sci.math Giuseppe Bilotta <bilotta78@hotpop.com> wrote:
> : Albert Wagner wrote:
> :> Look at your 'definition' restated, but without changing the
> :> meaning: A set S is said to be infinite if a proper subset of S
> :> is also infinite (in which case a bijection would exist).'
>
> : Wrong. The original definition makes no statement whatsoever
> : about the finiteness or not of the subset.
>
> : *You* are deducing that the subset is itself infinite
> : (something which, BTW, you can only deduce if you accept that a
> : set which is in bijective correspondance with another set has
> : the same finiteness property; where do you get this from?), and
> : putting it back in the definition.
>
> It is fairly easy to prove that if A is infinite, meaning
> that there exists a bijection between A and a proper subset
> of A, and that if there exists a bijection between A and some
> set B, then B must also be infinite. It is a consequence
> of the definitions, not part of the definitions.
>
> Let f be the mapping from A to B, and let f' be the
> mapping from B to A (ascii notation is rather limiting).
> Let g be the mapping from A to a proper subset.
>
> Then f(g(f'(x))) maps B to a proper subset of B.
>
> No recursion necessary.
>
> I would like to see an example of a definition as
> defined by the philosophers. Of course one of their
> rules seems to be that you cannot define by example,
> which seems to be a handy excuse for never having
> to actually demonstrate what they mean by anything.
>
> They various definitions proposed for "circle" have
> been very weak and supposed all sorts of other definitions
> such as "motion", "line", "rate" and required myriad
> assumptions about how those words are defined.
>
> Stephen
>
Circle: Two dimensional shape that maximizes area per perimeter. 

-- 
Smiles,
Tony

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