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Virgil

Posts: 4,480
Registered: 1/6/11

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

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In article
<c293798d-5b7e-4eba-a4df-6332461dc681@f19g2000vbv.googlegroups.com>,
WM <mueckenh@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.

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) 1-59]
> http://www.earlham.edu/~peters/writing/infinity.htm#galileo
>
> Regards, WM
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