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

From: Leonard Blackburn (blackbur_at_math.umn.edu)
Date: 07/01/04 

Date: 1 Jul 2004 08:22:06 -0700

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.

-Leonard

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