Re: how to list all of the real numbers

On Aug 10, 2:32 pm, lwal...@xxxxxxxxx wrote:
On Aug 9, 6:18 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Aug 9, 4:40 pm, lwal...@xxxxxxxxx wrote:
I agree that infinitesimals are possible, but unfortunately most
mathematicians will only consider standard theories.

Lots of mathematicians work in areas of specialty in which
infinitesimals are not relevent to their particular research. However,
just about anyone who works in foundations, including mathematicians
who work primarily in classical mathematics, knows about non-standard
models and of IST and of infinitesimal calculus on such bases. In
fact, non-standard models that provide for infinitesimal calculus are
PART of classical mathematics in the very sense that the existence of
non-standard models that provide for infinitesimal calculus is indeed
a theorem of ZFC and discussion of the matter is often even presented
in introductory textbooks in mathematical logic. Indeed non-standard
analyis is the result of a "stanadard" mathematician whose very GOAL
was to develop infinitesimal calculus within classical mathematical
logic.

The very fact that they are described as "nonstandard" implies that
they are not regarded as being standard theories at all.

You're making the mistake of getting hung up on how certain things are
NAMED while you are ignoring the SUBSTANTIVE information I'm giving
you: Non-standard analysis, in its model theoretic form, is a result
of the classical mathematics of ZFC (as the mathematical logic that
gives us non-standard models is easily formulated in ZFC). Theorems
that state such and such about non-standard models are theorems of
ZFC. And IST, the axiomatic approach to non-standard analysis, is a
conservative extension of ZFC, so ZFC is a subtheory of IST.

You keep saying things about "most mathematicians" and "standard
mathematicians" but you don't seem to be aware of the actual
literature of the subject or even that ordinary mathematical logic
presents non-standard analysis even as an INTRODUCTORY part of the
subject.

I have in fact heard of nonstandard analysis.

My point was not whether you'd heard of non-standard analysis, but
rather your apparently not knowing and understanding that it is a
result of classical mathematics and that it is not at all uncommon
that those you call "standard" mathematicians study non-standard
analysis, as I even mentioned one may find the subject of non-standard
analysis as part of even introductory textbooks in mathematical logic.

I do know that the
cornerstone of nonstandard analysis is the Transfer Principle axiom
(or schema). Some of the so-called "cranks" have actually made
statements that actually seem to accord with the Transfer Principle
in some way. (I first learned about the Transfer Principle by
reading about it in a thread on this newsgroup.)

Cranks barely have (even if at all) a coherent understanding of the
axiom of extensionality let alone something as advanced as the
transfer principle. And nothing cranks do is "in accord" with non-
standard analysis, since non-standard analysis ITSELF presumes
standard mathematics, as well as there is no "in accord" for people
who don't even have INFORMAL logical principles they can articulate,
let alone a formal theory.

The simplest example is that if for every natural number n there
exists a greater natural number, then the Transfer Principle states
that there exists a nonstandard natural number greater than all of
the standard natural numbers. In other words,

forall n(exists m(m > n)) implies (exists m(forall n(m > n))

Many so-called "cranks" have written posts similar to the above. But
as soon as one does, someone will immediately accuse the poster of
Quantifier Dyslexia, since apparently all the "crank" did was
reverse the existential and universal quantifiers. Indeed, many
applications of the Transfer Principle look like Quantifier
Dyslexia to the mathematician who is not using NSA.

Nope. Certain cranks babble incoherently which includes occasionally
placing references to non-standard analysis. Just that there are
certain things in non-standard analysis that sound similar to the word
salad of cranks does give the cranks any legitimacy at all.

I admit that many of the "cranks" don't even know what the Transfer
Principle even is, or that they are using it. The key point is that
many posters (including both Ross Finlayson and Tony Orlow who
regularly mention infinitesimals in their posts) would apparently
support a set theory that includes the Transfer Principle.

You don't know what YOU'RE TALKING ABOUT. Non-standard analysis ITSELF
presumes standard mathematics, either as model theoretically derived
or as ZFC is a subtheory of IST. (Of course, I'm not ruling out that
there are other approaches to non-standard analysis aside from
classical model theory and IST; but you can bet your sweeet bippy that
these cranks are not themselves aware of such advanced mathematics and
wouldn't have a clue as to how their prattle might COHERENTLY relate
to it.)

But the
standard mathematicians are much more likely to accuse them of
Quantifier Dyslexia than using the Transfer Principle. In other
words, illogic rather than an alternate logic.

Nope. You couldn't get these cranks to articulate an alternative logic
if you offered them a million dollars.

In other words, about half of the so-called "cranks" seem to
support ZF-Infinity+~Infinity and the other half seem to support
some form of Transfer. As you point out, none of them are
rigorous in their arguments, but to me these are the theories
which best approximate their beliefs.

So what? The chunk of gold that best approximates the wad of crumpled
paper in my wastebasket is about 1.5 inches in diamater. That doesn't
make the wad of crumpled paper in my wastebasket similar to a chunk of
gold in important aspect at all.

MoeBlee

.

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