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

From: Michael Barr (barr_at_barrs.org)
Date: 06/30/04 

Date: 30 Jun 2004 11:10:09 -0700

ddtl <this.is@invalid> wrote in message news:<tds4e0p4cuufic6va3val5m94n522nij7k@4ax.com>...
> >>and
> >> x'+y'=(x+y')'=((x+y)')'=(x'+y)'
> >
> >I'll let you translate this one...
>
> But this is not the point - what you have done, is translating the
> proof into a more modern language, but an underlying proof is still
> not valid. Sure, x'+y'=(x+y')' cat be written as f_[x'](y')=(f_x(y'))',
> but it doesn't change the fact that you you need associativity theorem
> in order to be able to perform such a transformation, and associativity
> theorem cannot be proved until you define what is addition and that
> an addition is possible!
>
> ddtl.

This can go around in circles for quite a while and get nowhere. Let
me start by saying at the outset that I don't care how careful Landau
was, the Peano axioms in this form are not sufficiently strong to
define any infinite subset or any function. Go look at them. Where
are functions mentioned? Subsets are, but only in the context that if
a subset is GIVEN with certain properties then it is all of N. Here
is the Peano axiom as given in modern topos theory (it starts with 0
instead of 1, but that is of no importance):

There is a recursive object N.

To explain this, I will use the following notation. 1 stands for the
one element set (terminal object of a category); if u:1 --> A is any
"element" of A and B is any other object, u_B is the composite of the
unique B --> 1 with u; composition of arrows g followed by f is f.g;
the identity on A is denoted A. For N to be recursive means that there
are maps 0: 1 --> N (a fancy way of saying that there is an element of
N we call 0) and a map succ: N --> N such that given any object A, any
function t_0: A --> N, and any map f: A x N --> N there is a unique
map t: A x N ---> N such that (A x t).(A,0_A) = (A,t_0) and (A x t).(A
x succ) = (p,f).t. Oh yes, (p,f): A x N --> A x N is the map whose
first coordinate is the projection on A and the second is f.

What this means in ordinary set theory is that given a map t_0: A -->
N and a map f: A x N --> N, there is a unique map t: A x N --> N such
that t.0_A = t_0 and f.t = t.(A x succ). Here is how that works for
defining addition. Take A = N, t_0 = N (the identity) and f(n,m) =
(n,m'). The first equation says that t(n,0) = n and the second that
t(n,m') = t(n,m)'. Then denote t(n,m) = n+m.

To do multiplication, let t_0 = 0_N and f(n,m) = (n.n+m), using the
previously defined +. Then the first equation says that t(n,0) = 0
and the second that t(n,m') = t(n,m) + n. And so on.

This leaves the question of why Landau and his successors thought that
what he had done was a proof. Interesting question, but I don't know
all the answers.