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


Re: Matheology � 176
Posted:
Dec 10, 2012 3:34 PM


In article <c293798d5b7e4ebaa4df6332461dc681@f19g2000vbv.googlegroups.com>, WM <mueckenh@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.
Intuition is an unreliable source of truth, as demonstrated by Cantor's prof that not all infinite sets are bijectable. > > 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.}}
Then go by injections which works well for both finite and infinite sets: Size(Set A) <= Size(Set B) if and only if A can be injected into B. And if Size(Set A) <= Size(Set B) and Size(Set B) <= Size(Set A) then Size(Set A) = Size(Set B) > > [Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2 > (1998) 159] > http://www.earlham.edu/~peters/writing/infinity.htm#galileo > > Regards, WM 



