On Jan 23, 3:47 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > ... > I take it that this new tack is so that you don't have to concede the > point: there is no mathematical publication which claims that the > above union contains elements larger than any natural, nor any > publication which claims that this is what it means to be infinite. > > If you want to discuss paths in trees, we must first finish this > point. > > -- > Jesse F. Hughes > > "Two years from now, spam will be solved." > -- Bill Gates, Jan 24, 2004
Usenet is a rather poor publishing medium, but Muckenheim won't be the first to suggest, in writing, that the natural numbers are a continuum and have infinite elements.
For example, in a theory where only finite sets are regular, the set of all of them contains itself: this is well-known as Russell's paradox.
As well where the hyperintegers model the integers, those are hyperfinite. This is in print.
That's not to discount Boucher's F nor Paris and Kirby's non-standard, countable, natural integers.