Dieudonne on Dedekind
arnold.neumaier at univie.ac.at
Wed Aug 11 02:27:43 EDT 2021
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.
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