Timeline for Why does the widespread erroneous definition of "irrational number" persist without being taught?
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| Sep 5, 2016 at 1:16 | comment | added | A.Ellett | @user52817 Dedekind cuts are built on ZF which has its own set of axioms. If the problem of a geometric proof is its reliance upon a particular postulate, then similarly for proofs using Dedekind cuts. Just as there are geometries (such as projective geometry) that don't rely on the parallel postulate, so too there are set theories that work with alternate axioms. The beauty of ZF is it points out a very nice way to build models for mathematics. But ZF isn't necessary. (You could approach math formalistically.) ZF is a model for much of mathematics; mathematics isn't necessarily ZF. | |
| Sep 5, 2016 at 1:01 | comment | added | user52817 | The geometrical proof that sqrt(2) exists is indeed the best you can do for high school students. But since it relies on the parallel postulate, by modern standards of rigor, it's weak. Since the premise of the question concerns defining the real numbers as analytic objects, I just want to point out this shortcoming. You did not say that d-cuts are antiquated, but you did say in reply to a earlier answer that they are not necessary. I am trying to point out that such an approach is necessary for a modern definition of the reals. | |
| Sep 4, 2016 at 23:47 | comment | added | A.Ellett | @user52817 i never said dedekind cuts etc were antiquated. ZF set theory (and is kin) are not necessary here. and i could not imagine trying to get high school students to understand them (there's a lot of very high level math involved with d cuts). but i would be interested to know why you believe a geometric proof is inadequate. | |
| Sep 4, 2016 at 23:39 | comment | added | user52817 | Working in a **model ** to establish existence. | |
| Sep 4, 2016 at 22:54 | comment | added | user52817 | The fact that the square root of 2 exists as a real number should eventually be established analytically, not geometrically. The standard Euclidean proof you cited using the diagonal of a square depends on the parallel postulate. So here you are working in a to establish existence. The analytic approach uses Dedekind cuts or Cauchy sequences. This is not an antiquated 19th century approach! | |
| Sep 4, 2016 at 20:49 | history | edited | A.Ellett | CC BY-SA 3.0 |
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| Sep 4, 2016 at 19:15 | comment | added | A.Ellett | @MichaelHardy I've taught in a variety of venues including at a small liberal arts college in the midwest, and most recently in an inner-city high school in a large metropolitan area. | |
| Sep 4, 2016 at 19:11 | comment | added | Michael Hardy | Where do you teach? | |
| Sep 4, 2016 at 18:55 | comment | added | A.Ellett | @MichaelHardy Regarding decimal expansions and diagonalization, it's an approach that has worked for me in the classroom. Diagonalization is relatively easy for the students to grasp, but I would never just leave it there. Hence, I always wrap up the sequence of lessons with the usual proof for the irrationality of $\sqrt{2}$. I'm not sure why you bring up Cantor or Cantor's proof. | |
| Sep 4, 2016 at 18:44 | comment | added | A.Ellett | @MichaelHardy Well put regarding the diagonal of the square. That's part of what I was trying to drive at. | |
| Sep 4, 2016 at 18:43 | comment | added | Michael Hardy | Moreover, the ratio of diagonal to side of a square is how the idea emerges naturally from geometry. | |
| Sep 4, 2016 at 18:43 | comment | added | Michael Hardy | I don't agree that using decimal expansions is the best way to show irrational numbers exist. In particular the diagonal argument is NOT how Cantor showed the uncountability of the reals. Look at this: en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article | |
| Sep 4, 2016 at 18:21 | history | answered | A.Ellett | CC BY-SA 3.0 |