Re: Rigorous proof of natural numbers' properties (by Edmund Landau).
From: Leonard Blackburn (blackbur_at_math.umn.edu)
Date: 06/30/04
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Date: 30 Jun 2004 10:06:46 -0700
David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<u635e095i8kvh2gl6i2i6ig50lphv5gked@4ax.com>...
> On Wed, 30 Jun 2004 10:01:39 +0200, ddtl <this.is@invalid> wrote:
>
<snip>
>
> Uh, that book has been around for a _long_ time (and Landau was
> an extremely careful writer to begin with). If you think one
> of the proofs is not valid you're missing something (perhaps
> not entirely your fault, probably it would be written in
> different language today).
<snip>
Be careful Professor Ullrich. Landau's book does indeed contain a fatal
error in the proof of the existence of an addition function. I admit I
only briefly skimmed this thread, but I can't help diving in right now
before reading it carefully, as this very issue arose for me two years
ago. At the University of Minnesota, Professor Max Jodeit had been giving
a course based in part on Landau's book (the one in question). He had
been teaching the addition theorem as Landau had presented it for some
time. I was his teaching assistant in 2002. I noticed the serious
error in the proof right after having been doing a lot of logic and
set theory including learning various recursion theorems (like the one
in H. Enderton's book _A Mathematical Introduction to Logic_. I pointed
out the error to Professor Jodeit, and he did not believe me for more
than a month of debate. But in the end he came to complete agreement with
me and he now teaches a correct proof based on a recursion theorem for
the natural numbers as given in H. Enderton's _Elements of Set Theory_.
In that book, Enderton even mentions that there are some erroneous proofs
of the addition theorem in print. Landau's is one of them. Enderton
mentions that if a proof does not utilize the Peano axioms that 0 is not
in the range of the successor function and that the successor function is
one-to-one, then the proof cannot be correct. See p. 76 in _Elements of
Set Theory_.
For my discussion of Landau's error go to
http://www.math.umn.edu/~jodeit/course/Math3283S02.html
which is Professor Jodeit's course page and click on this link near the
bottom:
"Leonard Blackburn's Notes and Comments on the Addition Theorem."
In this document, I've written everything but the first paragraph (written
by Professor Jodeit). I don't agree with the emphasis on the proof being
ok except for missing justification that is expressed in that paragraph.
The matter is also very briefly discussed in version 5 (?) of the Peano
Postulates links.
I hope this document also helps the OP.
Sincerely,
Leonard Blackburn
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