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Re: Cantor's first proof in DETAILS
Posted:
Dec 1, 2012 4:24 PM
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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 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.
>
> http://arxiv.org/abs/math/0106100
>
> 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":http://philpapers.org/rec/PARSSA-3
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.
Thanks, Fred.
I promoted that notion, proper subsets are lesser than their subsets,
and was roundly derided, until I pointed them to Katz' Ph.D, from
M.I.T. Now also I'll direct them to Euclid.
Were they wrong before, or after? Or, were they just ignorant? Were
they arguing the point, or the poster? Did they learn, or were they
taught? Are just fools, or just fooled? Does it matter anyways
because there's no use for transfinite cardinals? (And measure theory
is plainly countable and defined by our rules of the integral calculus
in the fundamental theorems of calculus, standardly.)
Then, with regards to the text, that's clear and rather plain, it's
readable, you get gibberish. Read what you read, this is a
statement. And that's technical, and correct.
So, whatever you read of it, here's some more.
Thank you, that's good reading and I enjoy it.
http://arxiv.org/abs/math/0106100
http://philpapers.org/rec/PARSSA-3
And then yes thank you I generally respect all and have a high respect
for Jeffries.
So, particle and wave: point?
Regards,
Ross Finlayson
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