On Mar 23, 2:44 pm, fom <fomJ...@nyms.net> wrote: > On 3/23/2013 4:34 PM, Ross A. Finlayson wrote: > > > > > In a sense, infinity _is_ the numbers. Start from even more > > fundamental objects than natural numbers as elements. Like the > > numbers, they are as different as they can be and as same as they can > > be, where they are each different in not being any other and each same > > in being defined by that difference. There's no stop to that, it's > > gone on, forever. Then, in a way like when you look into the void, it > > looks into you > > Is this your way of saying that if you look > into the void, you and the void become one? > > http://en.wikipedia.org/wiki/Cantor-Bernstein-Schroeder_theorem > > Just kiddding.... > > I think Cantor would appreciate your sentiment that > the numbers of Cantor's paradise are more fundamental > than those of Kronecker's torment.
I wouldn't say that infinity, even in the numbers, is either of those things. In ZF, Infinity is _axiomatized_ to be an inductive set, and a well-founded/regular one, that's not a given. Calling that the universe, Russell's comment is that it would contain itself.
There's a case for induction, as it were, that each case exists. Then it is to be of deduction, not fiat by axiomatization, from simple principles of constancy and variety, the continuum.
In a theory with sets as primary objects, a set theory and a pure set theory, numbers would be very rich objects indeed, as not just individual elements by their elements, but all relations of numbers. Set theory (well-founded, as it were, regular or that objects are transitively closed) is at once over-simplification, to talk about anything besides sets, and over-complexification, to talk about itself when any universal statement is in the meta.
There are no numbers in a pure set theory. To call the natural integers a set, it contains only numbers, for the Platonists: elements of the structure, of numbers, as: none exist in a void.