Date: Dec 2, 2012 4:12 PM
Subject: Re: Cantor's first proof in DETAILS

On Dec 1, 3:45 pm, Virgil <> wrote:
> In article
> <>,
>  "Ross A. Finlayson" <> wrote:

> > On Dec 1, 8:49 am, FredJeffries <> wrote:
> > > On Nov 30, 8:42 pm, "Ross A. Finlayson" <>
> > > 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 soi-disant 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 post-modern 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.

> > >
> > > See also Matthew W Parker "Set Size and the Part-Whole Principle"
> > > and references therein where he demonstrates that methods like Katz's
> > > "must be either very weak and narrow or largely arbitrary and
> > > misleading":

> > 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 part-whole 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 can t."

> > 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 non-existence 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.
> --

Valid EF is just the one function, standardly modeled by those finite
initial approximations with the integers bounded above.

How's that? (With "[Virgil's] truly random device assigning each one
to be even or to be odd".) The numbers are already even or odd for
whether they're congruent to zero modulo two, i.e., integral multiples
of two, or not (and if not, exactly and only multiples of halves).
That kind of ridiculous misrepresentation serves nothing but to
cheapen the discourse in misdirection from matters at hand. What
you'd reassign what "even" is, from the integers in order? Here
plainly an integer "at random" from between zero and a large bounded
quantity has that a reasonable estimate in as to whether it's even is
via the outcome of a fair coin toss. Now, this can lead to a
discussion on the central limit theorem and alternatives. There's
still much for the theory of probability to have developed in the
simple theory of attenuation.

Consider this: a bookmaker offers more than two for a wager of one on
correctly guessing whether a strongly pseudo-random integer is even.
Is not the game a winner with a constant guess of yes? Similarly for
correctly guessing whether the result is a multiple of n = three,
four, etcetera, with a return > n, do you not see that it's a winner?
Is not the bookmaker irrational and soon, starting with finite
resources, broke, fairly accepting wagers on those terms? (I'm a
capitalist not a gambler.) Half of the integers are even.

Funny, I respond to points with points and you remove them from your
replies, in a debate that would be seen as "dropping" the points. Go
with the flow. Perhaps you're familiar with rhetoric, and forensic
rhetoric? Go with the flow.

And please let it be known I happily and readily address mathematical
points, and do, and I don't drop points.

No you fool they're words. You write to bite? Write to enlighten
(and gallery, please understand that my goal is to promote
mathematical discourse). Quit gumming everybody, quit pushing your
pablum on those beyond it, and damnit quit snipping and yipping you
obnoxious semi-intellectual poltroon.

And that's generous.

Then, it was just established that the part-whole principle is
reasonable, for describing the relative properties of sets and their
proper subsets, in their supersets, but that is just peer-reviewed and
defended to the thesis board only recently, historically. It's
brought into the debate, and debated, showing that when it was
verboten, that was wrong.

Remarkably, I yet stand.

Transfinite cardinality: applied, where?

Warm and seasonal altruistic regards,

Ross Finlayson