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

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Posts: 1,968
Registered: 12/4/12
Re: Matheology § 176
Posted: Dec 11, 2012 12:33 AM
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On 12/10/2012 3:11 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.

To wit,

Definition of proper part:
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

Definition of membership by Aquinian individuation:
AxAy((xey <-> (
Az(ycz -> xez) /\
Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw))))))

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