Dieudonne on Dedekind

[Richard Grandy]

Motivated students may be intrigued by the hole.  But most of the average class will be turned off by Dedekind cuts, etc.

I was a math major and in a number theory course junior year the prof started with the Peano axioms and said that if you were interested in where
those came from you could read Russell.  I finished the math major but found the “where do the axioms come from” a more interesting long run question and
became a philosophy professor for 50 years.  Not the worst outcome, IMO.

> On Aug 11, 2021, at 1:27 AM, 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.
>