Date: Mar 21, 2014 3:51 PM
Subject: Re: � 454 Equality and the axioms of natural numbers
In article <email@example.com>,
> On Friday, 21 March 2014 09:30:12 UTC+1, Virgil wrote:
The question was: If the five truncated Peano axioms, not more and not
> > > less,
> > > define the natural numbers
> > But the "truncated"Peano postulates prove nothing.
> They prove that 1 (or 0) is a natural number.
My "truncated PA's read
There is a set, S, and an object, o, and a function, f, such that
1. o is a member of S
2. if x is in S then f(x) is in S
3. For every x in S, o =/= F(x)
4. For every x and y in S, if f(x) = f(y) then x = y
5. If a set T is such that o is a member of T and whenever x is a
member of T then also f(x) is a member of T then S is a subset of T.
Since that "o" is only specified as being an object and the postulates
never mention natural numbers, WM's assumption is, as usual, unfunded.