Dieudonne on Dedekind

[JOSEPH SHIPMAN]

“Gaping hole” seems a bit strong. “Downward closed sets of rationals” and “infinite sequences of decimal digits” both seem like straightforward concepts that students will not regard as baffling, and both the correspondence between these and the LUB property for each are close to the low end of the levels of difficulty students who do proofs must face.

It’s possible to present the Dedekind cut construction badly, but it’s still way easier than Cauchy sequences.

— JS

Sent from my iPhone

> On Aug 11, 2021, at 2:40 PM, Arnold Neumaier <Arnold.Neumaier at univie.ac.at> wrote:
> 
> 
>> On 11.08.21 04:24, Timothy Y. Chow wrote:
>> It's true that this is close to how professional mathematicians tend
>> to view the matter.  Dedekind cuts or Cauchy sequences are used to
>> prove existence, but once you've done that, you forget about the
>> existence proof and just use the axioms.
>> 
>> I'm not sure how I feel about this attitude.  It's certainly a
>> pragmatic one---you can cover more material this way, and the students
>> who don't see the point of the existence proof are spared the torture
>> of being dragged through it.  On the other hand, as a fan of f.o.m., I
>> can't help but feel some pangs of conscience about knowingly leaving a
>> gaping hole in the foundations of the subject, right around the time
>> that most students are just beginning to understand what rigorous
>> proof is.
> 
> A gaping hole in undergrad education is unavoidable. Dedekind's proof
> only moves the gaping hole from the foundations of real numbers to the
> foundations of set theory.
>