Virgil
Posts:
8,833
Registered:
1/6/11


Re: � 454 Equality and the axioms of natural numbers
Posted:
Mar 21, 2014 3:51 PM


In article <fe9e9fb1c93a430787da48cb1242957d@googlegroups.com>, mueckenh@rz.fhaugsburg.de wrote:
> 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 Peano Postulates: 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. 

