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12/4/12


Re: Matheology § 176
Posted:
Dec 11, 2012 12:33 AM


On 12/10/2012 3:11 PM, Virgil wrote: > In article > <c293798d5b7e4ebaa4df6332461dc681@f19g2000vbv.googlegroups.com>, > WM <mueckenh@rz.fhaugsburg.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 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 > > 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 selfnesting.
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))))))



