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YBM
Posts:
804
Registered:
11/27/09


Re: Matheology § 176
Posted:
Dec 10, 2012 6:47 AM


Le 10.12.2012 10:23, WM a écrit : > > > 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 > > Regards, WM
Only a disgusting dishonest crank like WM may have forgotten to quote the conclusion of this article:
> Properly understood, the idea of a completed infinity is no longer a problem in mathematics or philosophy. It is perfectly intelligible and coherent.



