Re: infinity
- From: "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx>
- Date: 14 Oct 2005 16:14:28 -0700
sci.math_20040113:
Those are things that come to mind when you think about infinity for a
couple years, particularly in the context of mathematical infinity and
even the state-of-the-art of set-theoretical mathematical infinity.
sci.math_20040113:
A lot of people have problems with the Cantorian proofs about the
uncountability of infinite sets, and I can see why: if you well-order
a set (eg the canonical ordering operator application) then
inductively, for each element, of one infinite set there one unique
element of the other.
sci.math_20040113:
The powerset result is kind of deep, to reconcile induction and the
reductio ad absurdum, I believe, or proof through contradiction, or the
powerset result, you get to quite a few avenues of the infinite as
zero, for example, or negative one, and the specific and specialized
identity of numbers and their application. For mathematicians, the
subfield of set theory particularly as it is accepted as the foundation
and explanation of the deepest roots of mathematics is to
mathematicians like the nature of god is to priests or the nature of
the universe to physicists, or the nature of being to humans.
sci.math_20040115:
That's not to say it's cut and dried, there are certain complexities of
dealing with the infinite.
sci.math_20040115:
It's certainly fair to say that of all the rationals, only
infinitesimally many of them are integers. There are infinitely many
integers, but for each, we can define a unit neighborhood around it
containing no other integers but containing infinitely many rational
numbers. The converse is not true.
sci.math_20040115:
The integers are a proper subset of the rationals, and as well,
infinitely many proper supersets of the integers are proper subsets of
the rationals. There are "more" rationals than integers for a variety
of reasons.
sci.math_20040115:
In talking about uncountability, what appears is the discussion of the
map between an infinite set and its powerset. In not considering the
universal set, which is its own powerset, we have to approaches to show
the infeasibility of mapping an infinite set to its powerset. Now,
don't get me wrong here, there is some mathematics that is quite valid
without this theorem and it can be axiomatized away quite brusquely and
integral calculus or non-standard analysis is unaffected. Anyways,
there are two basic approaches, the set theoretic one says that for any
function f(x) from a set X to its powerset, that one element of the
powerset, denoted S, is the set of all values where if not(x E f(x))
that x E S. S is supposed to be precluded from being in the range of
the function f in this way. In consideration of the naturals X as
ordinals and with f(x)=x+1, S={}, which is a subset of X and thus an
element of P(X). With f(x)=x, S=X, which is also a subset of X and
thus an element of P(X).
sci.math_20040115:
Here one idea I had about that was to consider instead of the powerset
a construction appelled the proper powerset, containing each proper
subset of the set. Then, for an infinite set, there is only one
element difference between the powerset P(X) and a proper powerset
PP(X), that being the element X E P(X), so basically there is one
element missing.
sci.math_20040115:
Another consideration of the powerset mapping is basically geometric in
tone, it says create a list of the infinite binary sequences with for
each index of the sequence that if the value is a 1 then the element
with that index's value is an element of that represented subset, else
it is not. Then, the antidiagonal is constructed which is different at
each i'th place in the antidiagonal from the i'th element of the i'th
sequence.
sci.math_20040115:
I think we'd all agree that any infinite set has a trivial mapping to
an infinite subset that is only disjoint one element, but we don't.That
two infinite sets can map ot each other used to be called "Galileo's
paradox", and now it's not. Very few integers are perfect squares, the
density of perfect squares among the integers is infinitesimal, and
characteristic of the density of the perfect squares within the
integers.
sci.math_20040115:
So anyways, "countability" and "uncountability" are words about
"cardinality", and specifically "the cardinality of infinite sets".
Can cardinality explain why the density of even numbers within the
integers is 1/2? No, it doesn't and plainly cannot as it is blind to
that distinction vis-a-vis "multiples of three", with density 1/3, or
other number-theoretic attributes of the infinite set of numbers.
sci.math_20040115:
Name one empirical, real-world result that hinges and rests upon the
cardinality of an infinite set.
.
- References:
- Re: infinity
- From: Jonathan Hoyle
- Re: infinity
- From: Virgil
- Re: infinity
- From: Ross A. Finlayson
- Re: infinity
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