Re: how to list all of the real numbers

On Aug 5, 10:16 pm, lwal...@xxxxxxxxx wrote:
....

Among the nonstandard mathematicians who do contradict
standard set theory, the most commonly challenged axiom is,
of course, the Axiom of Infinity. Although ZFC with Infinity
replaced by its negation is consistent if ZFC itself is, and
there is a branch of mathematics called "finitism," it is
the general belief of most standard mathematicians that
anyone who denies Infinity in the 21st century merits the
label "crank."

Even denying Axiom of Choice, considered controversial a few
short years ago, may possibly make the Axiom's opponent a
borderline "crank." I've seen a few threads challenging
Choice recently, and the standard mathematicians who reply
in such threads defend Choice fully.

There are only a few well-known axioms which one may safely
deny without being labeled a "crank." One such axiom is the
Continuum Hypothesis. An adherent of ZFC+CH is not a "crank,"
nor is an adherent of ZFC+not CH. Also, there are the various
large cardinal axioms, some of which are supported by
different axioms. But try denying AC, or especially Infinity,
and standard mathematicians will not hesitate to use the
"crank" label.

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
there exists an infinite set of all the finite sets that contains only
finite sets) is not true, thus that some other axiom or deductive
notion better represents a statement that there exists an infinite
collection.

No, not everyone who thinks Cantor is not the end-all be-all when it
comes to infinity is a crank. Some, like myself, for example, find
simple deductive arguments to show that the foundation for the
diagonal and powerset results is wrong, that is, framed in an
inconsistent theory, or ZF is inconsistent.

Nobody actually uses transfinite cardinals for much of anything. The
complete density of the continuum (real unit interval) vis-a-vis the
sparse stellated points of the integers, as continuum, find that
almost all useful bookkeeping of things infinite happen in analysis
where the integral (over x) between zero and one of a dx is equal to
one (in units of area).

The notion of limit as finitizing analysis finds in the induction of
the argument that the limit does not equal necessarily any value of an
expression for finite inputs of values, only infinite inputs. To
think that the limit (as for example found using the methods of
calculus) really, truly is the evaluation, that the sum of half and
halves again forever truly is equal to one, reflects in disbelief
absolute cessation of movement, Zeno stall, as it were.

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 think transfinite cardinals are not deep. Basically they say "one
infinity, two infinities, three infinities, more", and of those "one,
two, three, etcetera." That would be unremarkable except for the use
of central items in mathematics, the natural integers and real
numbers, in ways that preclude, in mathematics, other sensible (or if
not obviously sensible, deductively valid) notions about those items.
Look at the simple arithmetic of the cardinals as compared to all the
ways that arithmetic could be, for example adding two halves and
getting one instead of two.

(In ZFC the real numbers (as a set) are consistently equivalent to a
wide variety of cardinal numbers.)

Plainly, there are useful mathematics that not only have no use for
transfinite cardinals but also find that reliance on transfinite
cardinals, while helping organize various structural concerns in the
large, is not a true picture of those concerns.

In induction when the limit (of a non-constant convergent sequence) is
the value (of the sum, for example, of the partioned values) then the
infinite induction must have led to an infinite value of the input,
because no partial sum is the sum. That's where the sum is otherwise
greater than the parts, but only some infinitesimally small (small non-
zero) amount greater, that when the difference vanishes, in arguing
that the distance vanishes, Zeno's runner breaks the tape, and the
convergent series then actually reaches its limit, and as it was for
no finite term and it has, it is for an infinite term,

As a non-conventional viewpoint, to be more to the point people have
been trying to crack Cantor's results since they were discovered.
That is not to say that many haven't accepted the powerset and
diagonal arguments as true results (in the infinite). It is instead
that there are other perceived directions for making rigorous and
sound the foundations of analysis, which find contradiction in those
rrresults. Comprised of simple statements about the natural and real
numbers, it is not beyond reason to think that if the diagonal and
powerset arguments have flaws they are in the presuppositions about
those collections of objects, rather than using Cantorian results to
return to, or not, what are standard, in the sense of being used ably
as a perceived foundation of things infinite and infinitesimal to
assuage any guilt about not having rigorous and moreso sound
foundations of the readily applicable infinitesimal analysis.

If the limit is the sum, and for no finite input is the evaluation the
complete sum, and the evaluation is the complete sum, then there is an
infinite input, in the natural integers.

Ross

--
Finlayson Consulting

.

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