Re: Rigorous proof of natural numbers' properties (by Edmund Landau).

From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 07/01/04 

Date: 1 Jul 2004 12:38:09 -0500

In article <aa503d8.0407010721.aa5ec10@posting.google.com>,
Leonard Blackburn <blackbur@math.umn.edu> wrote:
>David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<bbo7e0lb7719hasc4k2v0dtuac06daj0eh@4ax.com>...

<snip>

>> ***Landau doesn't _say_ that there is a function defining
>> addition! He simply states that x + y "can be defined"
>> for all x and y.***

<snip>

>> Hmm. I bet it's possible to construe things so that
>> the gap exists only in things like my "translation"
>> of what he wrote.

<snip>

>You have given me much to think about. Unfortunately I am in the
>process of moving to a new dwelling and am quite busy. I may get back to
>this later. But above, I think you have an excellent point. Now that
>I recall, the proof I was attacking was Professor Jodeit's translation
>of Landau's proof in which Jodeit asserts the existence of an addition
>_function_. I hadn't even read Landau at the time.

>Also, I think in a sense, Landau does just have gaps rather than errors,
>but I think it is possible that the gaps are serious in this respect:
>If one fills in Landau's gaps, then one will have a proof that is
>unnecessarily long. The tools used to fill in the gaps could be used
>to give a much simpler argument.

>Also, I apologize for not carefully considering how any of this applies
>to the OP's orginal questions. You are probably correct that my comments
>don't apply.

The easiest way to correct the gaps in Landau's approach
for pedagogical purposes is to assume the existence of
addition and multiplication; once these are in place, the
rest follows.

In fact, I believe that we should teach the ordinal
structure of the integers in first grade, and later tie
the cardinal into it. These ARE different concepts. To
do this, I would start the integers with 0, although
this makes some things SLIGHTLY more complicated, and
add powers so the usual notation can be done. The
"definition" of addition in Landau states that "counting
on fingers" is correct, and that of multiplication
becomes repeated addition. 

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558

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