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Virgil
Posts:
8,833
Registered:
1/6/11


Re: Matheology � 176
Posted:
Dec 11, 2012 12:32 AM


In article <225e367822564cf2858a5a9ccdc70001@g7g2000pbi.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:
> On Dec 10, 1:11 pm, Virgil <vir...@ligriv.com> wrote: > > In article > > <c293798d5b7e4ebaa4df6332461dc...@f19g2000vbv.googlegroups.com>, > > > > > > > > > > > > > > > > > > > > WM <mueck...@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. 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 > > counterintuitive. 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." > > > > http://www.earlham.edu/~peters/writing/infinity.htm#galileo > > > >  > > > Baruch Spinoza, a 17th century enlightenment thinker, sees the natural > integers as a continuum, of individua (thanks Eco, for the word). > > http://en.wikipedia.org/wiki/Baruch_Spinoza > > 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. > If that's a quote, put it in quote marks, since otherwise it marks you as the wiseass little old lady who originated the phrase. 



