Re: how to list all of the real numbers

On Aug 11, 10:47 pm, lwal...@xxxxxxxxx wrote:
On Aug 11, 7:29 pm, Virgil <vir...@xxxxxxxxxxx> wrote:

Most mathematicians will consider systems of infinitesimals possible,
but unless their own work makes such infinitesimals useful, they are not
liable to spend much energy or time on them.

"Crank" #1: Why accept the Axiom of Infinity?
Standard Mathematician: Because it is useful.
"Crank" #2: Why reject infinitesimals?
Standard Mathematician: Because they are not useful.

A mathematician is not "rejecting" non-standard analysis just by the
fact that the mathematician doesn't choose to do work in non-standard
analysis any more than a scholar of literature is "rejecting" Faulkner
just by choosing to study Chaucer instead. In other words, it's not a
"rejection" in the sense of a claim that something is generally
unworthy or incorrect or even necessarily of less merit.

So far in both this thread and in "An Inconvenient Truth" the
consensus
among standard mathematicians is that the infinite sets are "useful,"
(in response to HdB's claims that the Axiom of Infinity is not useful)
while infinitesimals are not "useful," which is why the former is part
of standard (or classical) analysis while the latter is relegated to
Nonstandard Analysis.

Infinitesimals are useful if you want to do that kind of analysis, but
since real analysis can be done without infinitesimals, to the extent
that one's interest is just in using the results of real analysis or
in working in ordinary real analysis, it's not necessary to also study
non-standard analysis. If my objective is to get across town, then a
taxi suffices; I'm not "rejecting" buses or limosines just by my
estimation that, for me, taxis are just fine.

The difference? The best argument I've seen is that it's not the
infinite sets themselves that are "useful," but the complete ordered
field that is important to the solution of differential equations.

You don't have to go to something as advanced as differential
equations to see the usefulness of infinite sets.

It
was stated that a complete ordered field cannot contain
infinitesimals,
for the set of infinitesimals are a set with no least upper bound. In
NSA differential equations become difference equations (with
infinitesimal differences), while integrals become sums of infinitely
many infinitesimals. And since differential equations and integrals
are easier to work with than (even finite) difference equations and
summations, infinitesimals are to be rejected. But one must assume
the Axiom of Infinity in order to prove that a complete ordered field
even exists.

One requires the axiom of infinity to prove the existence of a
complete ordered field from the rest of the axioms of SET THEORY. But,
anyway, what point are you suggesting in this regard?

MoeBlee


.

Relevant Pages

  • Re: how to list all of the real numbers
    ... but unless their own work makes such infinitesimals useful, ... field that is important to the solution of differential equations. ... further some year and so ago in "On well-orderingof the reals, ... Then, where a real may be a mark of a point, consider whether Hardy ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... but unless their own work makes such infinitesimals useful, ... among standard mathematicians is that the infinite sets are "useful," ... field that is important to the solution of differential equations. ... the Axiom of Infinity in order to prove that a complete ordered field ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... but unless their own work makes such infinitesimals useful, ... Why accept the Axiom of Infinity? ... among standard mathematicians is that the infinite sets are "useful," ... field that is important to the solution of differential equations. ...
    (sci.math)
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