
Re: The uncountability infinite binary tree.
Posted:
Dec 16, 2012 7:42 PM


On Dec 16, 2:56 pm, Virgil <vir...@ligriv.com> wrote: > ... > > You have claimed it, but never established it, and cannot do so until > you can DISPROVE the theorem that sets that the cardinality of any set > is less than that of its power set. > > ...
Alternatively, he could be proving that the elements of the reals are sensitive to their order, they're orderingsensitive.
For example that could include the consideration of the continuum of reals as at once complete ordered field and ring of iotavalues. That's not inaccessible to anyone who's ever even wondered why .999... equals 1, or not. That includes, for example, Newton and Leibniz, inventors of the integral calculus, where Leibniz' intuitive notation for summation is ubiquitous.
In a set theory with a universal set, it's its own powerset. In a theory with a Universe: as a set, it's its own powerset, that is an empirical rationale, that the mathematics founding it are so.
Skolemize, N^G contains no elements not in N, consider Hausdorff's apocryphal uncountable countable union of countables.
In ZF, with N as ordinals, in consideration of Cantor's settheoretic powerset result: with f(x) = x+1, S = {}, with f(x) = x, S = N.
With f(x) = x+1, there are no elements in S, where empty is an ordinal. For some, the direct sum of infinitely many copies of N is empty.
For the empty set as ordinal, or lack thereof, with f(x) = x+1, S = {}, with f(x)=x, S={}.
In ZF the empty set is defined as a subset of any/every set. But, it has no elements: it's not defined by its no elements. It's defined as a constant, along with the other constant so defined, an inductive set or N. So there are no derived elements of ZF in S, not in the range of f(n).
As ordinals, powerset is successor (for its elements) is order type. Per Katz, proper subsets are demonstrably smaller than their sets.
http://groups.google.com/group/sci.math/msg/682e6b2aa0c7a0ea?hl=en http://groups.google.com/group/sci.math/msg/9185fa16d50afd0b?hl=en http://groups.google.com/group/sci.math/msg/26b4b74530aeed18?hl=en
Then, where that the universe would be its own powerset is strong enough rationale to consider the case, the consideration of theories extra ZF, with regards to the defined or urelements of ZF, gets then to that it is just the one element, for all.
And, that has a strong foundation in the technical, in the philosophy: Leibniz' monad, Kant's DinganSich, Hegel's and Heidegger's Being/Nothing: Janus' introspection, the dialetheic and paraconsistent urelement of the null axiom theory.
Universe: own powerset.
Regards,
Ross Finlayson

