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

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Posts: 8,833
Registered: 1/6/11
Re: Matheology � 176
Posted: Dec 10, 2012 4:11 PM
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In article
WM <> 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]
> Regards, WM

Note that the part in {{ }} above is WM's addition, which runs totally
counter to the Peter Suber's own conclusion which reads:

Properly understood, the idea of a completed infinity is no longer a
problem in mathematics or philosophy. It is perfectly intelligible and
coherent. Perhaps it cannot be imagined but it can be conceived; it is
not reserved for infinite omniscience, but knowable by finite humanity;
it may contradict intuition, but it does not contradict itself. To
conceive it adequately we need not enumerate or visualize infinitely
many objects, but merely understand self-nesting. We have an actual,
positive idea of it, or at least with training we can have one; we are
not limited to the idea of finitude and its negation. In fact, it is at
least as plausible to think that we understand finitude as the negation
of infinitude as the other way around. The world of the infinite is not
barred to exploration by the equivalent of sea monsters and tempests; it
is barred by the equivalent of motion sickness. The world of the
infinite is already open for exploration, but to embark we must unlearn
our finitistic intuitions which instill fear and confusion by making
some consistent and demonstrable results about the infinite literally
counter-intuitive. Exploration itself will create an alternative set of
intuitions which make us more susceptible to the feeling which Kant
called the sublime. Longer acquaintance will confirm Spinoza's
conclusion that the secret of joy is to love something infinite."

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