Re: how to list all of the real numbers

On Aug 9, 4:40 pm, lwal...@xxxxxxxxx wrote:
On Aug 9, 9:24 am, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

Denying the Axiom of Infinity doesn't necessarily make on a finitist:
instead perhaps there is some consideration that the characterization
of an infinite set by the axiom of infinity as generally stated (that

I see what you mean. In other words, what you are suggesting is that
one replace the Axiom of Infinity with another axiom phi, such that
ZF-Infinity+phi proves the existence of a set that is not hereditarily
finite, but one can prove certain theorems that match one's intuition
yet whose negations are provable in ZFC.

No, not everyone who thinks Cantor is not the end-all be-all when it
comes to infinity is a crank.

But standard mathematicians, of course, believe otherwise.

If a function bijects between the natural integers and reals then it
shares a variety of properties with the Equivalency Function, which
maps the first integer after zero to the first real number after zero
in the total linear and well-ordering there of the reals, the natural
order. That the value EF(1) would be not only an infinitesimal,
hinting that the real numbers contain infinitesimals, but furthermore
a value nearer zero than any other value, contradicting the density of
the reals within themselves as always having infinitely many between
any two, leads to some non-standard notions of the real numbers.I wouldn't characterize the notion of rejection of the Axiom of Infinity that way, because I think a true specification or "axiomatization" of infinity would see there a contradiction with the Axiom of Regularity, if for example infinite sets were irregular.

So, for example, instead of "ZF - AoI + phi", for some alternate
formulation phi of the specification of the existence of an infinite
set, it would be more along the lines of "ZF - AoI - AoR + phi", for
some phi expressing that alternate specification of the existence of
an infinite set and perhaps anti-foundation.

With regards to the transfer principle, you might consider Schmieden
and Laugwitz formulation (or rather, model) of the real numbers where
besides being the complete ordered field, in terms of various
operations upon real numbers, that the set that comprises the
continuum of the real numbers has as well structure as a "partially
ordered ring with rather restricted transfer principle", as I believe
was coined to express their conception of the real numbers. That is
to say, as I have described here in some detail, there is a
consideration that given various operations on the real numbers, they
form the complete ordered field, but as well there is another set of
operations, basically able to find successor reals or viewing real
numbers as a contiguous sequence of points.

(The antidiagonal argument doesn't work in unary, binary, trinary, or
"base infinity". The unit impulse function is not a real function. )

With regards to the notions of infinitesimals and where they exist the
notion that there are infinitesimals somewhere among the reals, the
reals being gapless thus there being infinitesimals in the reals, thus
that some real numbers have properties that do not meet the standard
specifications of the reals numbers, that leads to a consideration
that the Dedekind/Cauchy method of expressing real numbers, where
Dedekind's is basically equivalent to I believe Eudoxus' construction
some 2000 years previous and equivalent to Cauchy's, that D/C cuts/
sequences are not adequate to represent the real numbers. There are
only real numbers between zero and one, if there are infinitesimals
between zero and one they're real numbers, elements of the continuum.

(There exist counterexamples in standard real analysis showing a least
positive real.)

I think most mathematicians don't bother creating, or assimilating, an
entire theory of mathematics, with the intuitionistic notions of given
truisms that are plenty upon which to build useful, pragmatic
results. It's not necessary for them that they do, besides, much of
foundations is specialist and technical, expressly and tediously
verbose in formalization.

Ross

--
Finlayson Consulting


I agree that infinitesimals are possible, but unfortunately most
mathematicians will only consider standard theories.


.

Relevant Pages

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