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


Re: Cantor's first proof in DETAILS
Posted:
Dec 1, 2012 6:45 PM


In article <f11ecbb193b24eb39c46539add9404ea@nl3g2000pbc.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:
> On Dec 1, 8:49 am, FredJeffries <fredjeffr...@gmail.com> wrote: > > On Nov 30, 8:42 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> > > wrote: > > > > > > > > > No, I only have a Bachelor's of Science degree (in mathematics thank > > > you and I know computer science). The guy who wrote a dissertation to > > > convince soidisant set theorists that half the integers are even has > > > a Ph.D. from M.I.T. He got it for writing a dissertation in set > > > theory that half of the integers are even. > > > > I have no time for nor interest in your faux postmodern gibberish. I > > only wish to point our that Fred Katz's dissertation did NOT show that > > "half the integers are even". The interesting thing about the paper > > was that he was unable to determine whether There are "the same > > number" of even natural numbers as odd or whether there is one more > > even than odd. > > > > http://arxiv.org/abs/math/0106100 > > > > See also Matthew W Parker "Set Size and the PartWhole Principle" > > and references therein where he demonstrates that methods like Katz's > > "must be either very weak and narrow or largely arbitrary and > > misleading":http://philpapers.org/rec/PARSSA3 > > > Well great, sure then that goes back to our discussion of whether > "exactly half the integers are even" or in as to whether "give or take > one element, of the infinitely many, exactly half of the integers are > even, and given an arbitrary method of selecting an integer with no > prior expectations, it is as or more reasonable than any other course > that the estimate is that half of those selected would be even." > > Simply, half of the integers are even. > > It would seem there would be more use of the modern concrete > mathematics, in number theory and in asymptotics, to constructively > build the partwhole principle than to aver that it's Euclid's and not > explored in recent history. > > Parker: "And while the _exploration_ of [...] theories of size is > certainly enlightening, one of the things it ultimately reveals is how > limited [...] sizes themselves are by their unavoidable arbitrariness. > This is not to say they are useless altogether; as noted they may well > have special applications to probability and number theory. However, > anyone who has hoped for a revolutionary new [...] theory of set size > with breadth and informativeness approaching what we would expect from > a notion of _how many_ will ultimately be disappointed." > > Concrete mathematics has applications of this simply today, and indeed > ready applications abound. Finite combinatorics is complete, and: > unbounded. > > "One might object that language always involves haphazard > stipulations, but this is > mainly in the choice of symbols used. Where the concepts expressed are > also somewhat arbitrary (like the culinary distinction between fruits > and vegetables, for example), this again limits their usefulness and > scientific interest." > > Then, Parker goes about describing that sets, as their elements are > plotted on their supersets with regular structure as they are > constructed, have simple translations where the "same" elements have > different sizes as a collection, but those aren't exactly the "same" > elements, _in the context of their being in the overall context_. For > example, shifting or translating the integers from the origin one > right, leaves not the same set of integers, for what they are objects, > defining the set by its elements. > > "So unless we want the size of a set to depend on its particular > position (even while holding the relative positions of the elements > fixed), or perhaps on the bare haecceities of its elements (if there > are such things), we should like ATI to hold. And for Euclidean sizes > it can¹t." > > It can't. So, Schnirelmann and number theory give us that half of the > integers are even, and Katz' result is that proper subsets are > demonstrably smaller than their supersets, _in all their supersets_, > in modern theory, fine. Thank you for clearing up that his direct > statement is to that effect, and that the even integers soundly have a > size less than that of all the integers, in the integers, follows from > further development. Here, it supports that no other rational number, > than one half, best describes how many of the integers are even.
If one went through the naturals, or the integers, with a truly random device assigning each one to be even or to be odd, then the expected deviation from half and half as the number assigned increases is also strictly increasing.
Of course if both subsets are infinite, they will have the same cardinality, but such infinite sets have the same cardinality as some of their proper subsets, so we cannot unambiguously speak of the "same number of elements" in a set and any of its proper subsets. > > I promoted that notion, proper subsets are lesser than their subsets, > and was roundly derided.
Proper subsets are necessarily inclusions of their supersets, but need not be of any smaller cardinality.
Cardinality is based on the existence or nonexistence of injective mappings. So that it is only for finite sets that a proper subset need be of smaller cardinality than its proper superset.
In fact this is one acceptable way to distinguish finite sets from infinite sets: A set is finite if and only if every proper subset of it is of smaller cardinality. 

