On 3/23/2013 5:09 PM, Ross A. Finlayson wrote: > 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. >
It is odd. In some sense, modern mathematics actually treats its objects as urelements relative to set theory. Looking at Hilbert, he makes statements whereby his formalism is intended to supersede the class-based constructions of Dedekind.
Your frank statement that a set is not a number reflects that sentiment.