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

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Posts: 2,720
Registered: 2/15/09
Re: Matheology § 176
Posted: Dec 10, 2012 11:52 PM
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On Dec 10, 1:11 pm, Virgil <> wrote:
> 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:
> "Conclusion
> 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."
> --

Baruch Spinoza, a 17th century enlightenment thinker, sees the natural
integers as a continuum, of individua (thanks Eco, for the word).

That's quite a fine quote, Suber's. Yet, we might not need the
infinite regress when reason gives us already the firmament, of the
continuum, as the individua, what it is.

And it's turtles: all the way down.


Ross Finlayson

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