On Mar 23, 3:34 pm, fom <fomJ...@nyms.net> wrote: > 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.
Particular finite sets are called ordinals, set-theoretic operations on them are defined that give the same results as Presburger/Peano arithmetic of the natural integers. The negative integers aren't simply the complement as in finite-word-width machine arithmetic, but again simple enough operations on sets (with the only ur-element being the empty set) give a "model" of the integers. Rationals are defined simply enough as equivalence classes over any pairs of integers, besides zeros, the reals then see the Least Upper Bound as axiom. These are all to match number-theoretic features, and largely suffice for integers and rational numbers, but not so obviously do sets suffice to represent thusly elements (and all of) the continuum of real numbers.
Then, though, to call the empty set the number zero: wouldn't that be the number zero wherever there's an empty set? Building upwards to have particular sets for each of of the finite integers: then to build the numbers as sets, is to build all the relations of the numbers as sets, not just as to a set-theoretic model of only that set of numbers' operations: but of all instances, besides the schema. Where the ur-element is any thing, it so implies all other things, and is so implied. The collection and aggregates of sets or categorization or refinement of types or partition or bounding of division, are all of the same corpus.
Here back to the questions as above:
1) is not forcing simply transfinite Dirichlet box? 2) are there any results due transfinite cardinals, not of transfinite cardinals? 3) is not an irregular model of ZF non-well-founded? 4) does not a model of ZF contain itself? 5) is ZF not a model of itself?