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Virgil
Posts:
4,483
Registered:
1/6/11
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Re: Matheology � 176
Posted:
Dec 10, 2012 3:41 PM
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In article
<1a842415-d278-4aae-bea8-e688bc5b856e@x10g2000yqx.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 10 Dez., 16:30, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> > On Mon, 10 Dec 2012 01:23:07 -0800 (PST), WM
> >
> >
> >
> >
> >
> > <mueck...@rz.fh-augsburg.de> wrote:
> >
> > >Matheology 176
> >
> > >Here's a paradox of infinity noticed by Galileo in 1638. It seems that
> > >the even numbers are as numerous as the evens and the odds put
> > >together. Why? Because they can be put into one-to-one correspondence.
> > >The evens and odds put together are called the natural numbers. The
> > >first even number and the first natural number can be paired; the
> > >second even and the second natural can be paired, and so on. When two
> > >finite sets can be put into one-to-one correspondence in this way,
> > >they always have the same number of members.
> >
> > >Supporting this conclusion from another direction is our intuition
> > >that "infinity is infinity", or that all infinite sets are the same
> > >size. If we can speak of infinite sets as having some number of
> > >members, then this intuition tells us that all infinite sets have the
> > >same number of members.
> >
> > >Galileo's paradox is paradoxical because this intuitive view that the
> > >two sets are the same size violates another intuition which is just as
> > >strong {{and as justified! If it is possible to put two sets A and B
> > >in bijection but also to put A in bijection with a proper subset of B
> > >and to put B in bijection with a proper subset of A, then it is insane
> > >to judge the first bijection as more valid than the others and to talk
> > >about equinumerousity of A and B.}}
> >
> > >[Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2
> > >(1998) 1-59]
> > >http://www.earlham.edu/~peters/writing/infinity.htm#galileo
> >
> > You know this is simply dishonest. The part above inside the
> > {{}} is not part of what Suber wrote, and isn't even a summary
> > of anything in that article. Giving a quote and _modifying_
> > it this way, without making it clear that what you added
> > was added by you, is simply lying.
> >
> > Why do you do that?
>
> The parts in {{}} always give my comments, as the regular readers know
> and as you have easily recognized too. It is not difficult to see, and
> I chose it because practically no author uses this kind of marking. My
> comments are necessary to put everything in right perspective.
WM judges it insane to argue that a <=b AND b <=a implies a = b.
Which is how set theory works things.
Id defines cardinality so that Card(Set A) <= Card(Set B) if and only if
A injects into B, so cardinality is a conseqeunce of injectability.
This definition causes no problems at all to anyone other than kooks
like WM.
--
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