[FOM] General Foundations Discussion
Harvey Friedman
friedman at math.ohio-state.edu
Sun Jul 27 18:40:00 EDT 2003
Reply to General Foundations Discussion
On 7/27/03 2:01 PM, "Ayan Mahalanobis" <amah8857 at brain.math.fau.edu> wrote:
> I don't think you tried to answer my question. Though I am thankful for
> your comments. Since you bought up the topic that in classical
> mathematics THE real numbers are defined explicitly and you put an
> emphasis to the fact that real numbers should be explicitly defined and
> be unique. Do the phrase "explicit definition" has any foundational
> value without trying to understand what it means or what it should mean?
"Explicit definition" is among the few richest and critical foundational
notions we work with.
> The constructions of sets which depend on the axiom of choice might not
> be explicitly defined for some and probably they are right in the
> context of Banach Taraski paradox and others.
E.g., there is no explicit definition in the usual language of set theory
which, provably in ZFC, defines a well ordering of the real line.
>
> My point being that if we go in the cauchy definition and use
> convergence to define the complete field of Reals then we are already in
> the trap of epsilon-delta to which I already raised an objection.
Epsilson-delta is not a trap, but rather an essential feature of mathematics
for well over a hundred years.
>
> In Errett Bishop's constructive mathematics cauchy reals are not
> isomorphic to Dedekind reals but they seem to be doing fine. So I have
> little difficulty in believing that the uniqueness is a big issue. I
> once tried to read through the work of Bell , "Infinitesimal analysis"
> and yes I also thought that some form of existence of those real numbers
> would be nice. Could some one more knowledgeable in this subject please
> contribute to this.
>
Uniqueness is a critical issue. In constructive set theory, one can also
prove that there is exactly one Cauchy complete ordered field, up to
isomorphism, provided one sensibly defines the notion in standard ways.
DIGRESSION. One variant of this uniqueness theorem requires the use of a
very weak form of sequential choice to prove this (a form that is provable
without the AxC if read in classical set theory). Other variants do not
require this. In any case, sequential choice is generally considered a
standard axiom of constructive mathematics.
For the purposes of constructive analysis (e.g., in the sense of Bishop),
only Cauchy completeness is worth anything - Dedekind completeness is not.
Harvey Friedman
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