In article <email@example.com>, firstname.lastname@example.org wrote:
> > The Peano axioms describe structures that are embedded in EVERY infinite > > set.
Not unless every infinite set contains a well-ordered subset, and it is not obvious that every infinite set must contain an ordered subset, much less a well-ordred one. > > For matheologians who believe in uncountable sets, you should say more > precisely: every countably infinite set.
Not at all, unless in WM's world there can be infinite sets which do not have countably infinite subsets. Whish, considering the manifold self-contradictions that already occur in WM's wild weird world of WMytheology, would not be at all surprizing.
> That is correct. The set of natural > numbers however is not every countable set but a very special countable set.
Does WM claim that there are infinite sets in which no subset can be order-isomprphic to the set of naturals?
> The Peano axioms fail to produce this set because they are too general. They > produce the set 1, S1, SS1, ...
Which works fine as long as each one differs from all its predecessors.
Any set that is order-isomorphic to a proper set of naturals would serve equally well.
> Some of the many sets produced are > 1, 1s, 1ss, ... Ambiguous unless one is guaranteed that they are all different > 1, 11, 111, ... > 1, 1^1, 1^1^1, ...
Unless WM is using "^" for something other than exponentiation, one has 1 = 1^1 = 1^1^1, ... , so WM is claiming a one-element set of all naturals. --