Unextl

Unextl is a verse that contains nests of universes, and verses containing universe, and so on to such a degree that even the von-Neumann Universe fails to characterize the size of any meaningful portion of this verse. To show just
how horrifically the von-Neumann Universe fails to characterize it, P(V), P(P(V)), Pgoogol(V), and even Pleast inaccessible cardinal(V) aren't even large enough to be considered peanuts to U(1). To get to U(1) you would have to construct V, but starting with V itself instead of the empty set... assuming the minimum possible size of U(1). To construct U(2), you would have to begin with U(1), but do not be fooled... the amount of times that powerset has to be applied now allows ordinals within the universe of U(1), which are so large they cannot even fit into a set. Unextl goes far, far, beyond even U(ord), using the universe of all U(\(alpha\)) to give the allowed ordinals.
== Definitions ==
To describe the size of Unextl, it is necessary to define some terms.
=== Definition 0.1 ===
Define (or rather axiomatize) U(n) as follows,
* \(\forall n, U(n)\models ZFCH_L\) where ZFCHL is as defined in LNGN
* \(\forall a\forall b<a, U^{U(a)}(b)=U(b)\) where UU(a) is the interpretation of U in U(a)
* \(\forall a\exists b, U(a) = V_b\)
=== Definition 0.2 ===
Define the class U as \(\bigcup _{\alpha} U(\alpha)\)
=== Definition 1.1 ===
Define a map \(S \rightarrow f(S)\) as follows
* If S={}, f(S) = a Universe
* Else, f(S) is a verse containing ord of every element of S
== Unextl ==
Unextl, then, is f(U). To comprehend the size of this, we can start with the first order universe of set theory, V. To construct it, we start by calling V0 the empty set, V1 the set of all subsets of V0 or P(V0), which is {{}}, and in general Vn+1 = P(Vn). At transfinite indicies, Vn is the union of all Vx for x<n. V, without subscripts, is the union of all Vn. There is a problem though, what is the domain for the numbers? Well, the universe V itself, of course. While this may not seem sensical, there is a point at which sets can no longer get any larger, and that is where V must stop, and V will be a proper class containing all sets that can be created like that. As you may be able to imagine, V is MASSIVE, as V is the class of all well-founded sets, and as previously mentioned, it is too big to be a set. 
Did you think we would stop there? V is only the universe of first-order set theory, what about second order? Well, second order set theory allows the creation of predicates, although predicates can act equivalently to classes containing all objects where the predicate is true, and so the class of all subclasses of V can act as the universe of second order set theory... wait a second! That's just the powerset. P(V) is the universe of SOST, and is much more powerful than just V. We can actually take the powerset of that to get third order set theory, some more powersets for \(\omega\)th order set theory... hmph, it would probably be better to just create V but starting with V instead of the empty set. But wait... this time we have a larger universe for even more ordinals as input to Vn!
Thus, we arrive at the U() function. Although defined differently, U(0) acts similarly to V, U(1) acts similarly to our augmented V starting from regular V, and U(2) is like if we did that yet again, with the even larger universe. In fact, the difference in the definition of U() allows us to use transfinite numbers, even ones without FSS, instead of relying on induction like our augmented V. And so, we can define U in a similar way to how we defined V, by being the union of all U(n), where n is allowed to be any number within U itself. Finally, Unextl can be described by assigning the universe to {}, assigning the verse containing infinitely many of each element of the class C to the class C. Finally Unextl is the corresponding verse to the class U.
Category:Multiverse