Re: Uncountable sets in CZF?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 09/03/04 

Date: 3 Sep 2004 09:13:12 -0700

Hi,

David, I want to compliment you, those are some very informative posts
and that post should be recommended reading. There are notions of
theories where numbers are primary objects in the theory and are not
sets.

Keith presented a statement that he could map a proper subset of the
naturals bijectively to the reals. What's the deal with that? That
reminds me that the direct sum of infinitely many copies of N is
empty, but not, by exceptional definition.

Then again having {} as an element of a set implies regularity by many
naive constructions of that predicate.

I think the ur-element is zero, and sometimes infinity, and one.

We've probably all had calculus instruction, or at least the
pre-calculus instruction about limits and the definition of a
derivative and particularly in regards to this context the definition
of the antiderivative, the integral. Familiarity with a strict and
perhaps overly strict concept of limit is a given, where that is a
relatively modern response to the vagaries of infinitesimals of Newton
and Leibniz' functional, useful, empirical result driving
infinitesimal analysis: the integral calculus.

An even more modern response arose last century in the consideration
of the infinitesimals and the establishment of definitions to allow
the consideration of hyperreals. The hyperreals as a set contain no
elements that are not elements of the reals, and they do include
infinitesimals.

Now before everyone hauls out umpteen proofs that 0.999... = 1, they
are perhaps true in a similar manner as to how some geometrical
results are true (sound, valid, correct), in the Euclidean geometry,
and not true.

McAnally, I don't know how long you lurked on sci.math before
beginning your voluminous well-informed postings, you may well know
that I have since shortly after discovering this unmediated forum
argued the notion that infinite sets are equivalent, ignorantly,
naively. My response to the declaration that there were not mappings
from the naturals to the reals was the definition of the Natural/Unit
Equivalency Function. It's defined in a way that the domain is the
naturals and the range is the unit interval of reals.

EF(0) = 0
lim EF(oo) = 1
EF(n+1) > EF(n)
lim (EF(n+1)-EF(n)) = 0

People ask then "what is EF(1)" and I say "iota" and they say "what is
(EF(0)+EF(1))/2" and I say "undefined" and then (EF(0)+EF(2))/2 =
EF(1) and then (EF(0)+EF(n))/m is defined only when m divides into n.

Anyways that leads to the consideration of iota, the least positive
real infinitesimal, with properties of an x-transcendental real, and
the "vague fugue".

Claims are made that no function bijects between the naturals and
reals for a) the antidiagonal argument, b) the nested intervals
argument, and c) Cantor-Bernstein transitivity. The antidiagonal
argument and nested intervals argument are shown to not apply, for
dual representation and enforced list order, and for functions with
properties similar to EF in monotonically mapping.

The Cantor-Schroeder-Bernstein theorem(s), about surjection either way
implying a bijection, presents an inconsistency in mapping between the
reals and naturals and reals and powerset of the naturals. To that
end I described a model (obviously part of the theory in disguise) of
ubiquitous ordinals or naturals. Another alternate consideration even
to that has the ur-element represented as zero and infinity. In the
model of ubiquitous ordinals, the order type is the successor is the
powerset, and a a set X and P(X) f(P(X)) = f(X)+1, and f(x)=x+1 for x
E X is a bijective mapping, with S = {}, which while being zero when
zero={} is also the ur-element and equal to X+1.

About "as looking from outside", metatheories/metamodels, extensions,
etcetera, you have in them a set N, the set of all naturals integers,
in one of those models mapping bijectively to a set R, the set of all
real numbers.

Regards,

Ross F.

Relevant Pages

  • Re: Calculus XOR Probability
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  • Re: Zenkins paper on Cantor
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  • Re: Zenkins paper on Cantor
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    (sci.math)
  • Re: induction vs Cantor
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