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Topic:
Matheology § 176
Replies:
10
Last Post:
Dec 11, 2012 12:33 AM



Virgil
Posts:
8,833
Registered:
1/6/11


Re: Matheology � 176
Posted:
Dec 10, 2012 3:41 PM


In article <1a842415d2784aaebea8e688bc5b856e@x10g2000yqx.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.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.fhaugsburg.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 onetoone 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 onetoone 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) 159] > > >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. 



