On Mar 10, 5:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > So isn't forcing simply Dirichlet box / pigeonhole principle? Without > appeal to forcing, simply define arguments in symmetry then as to > transfinite pigeonhole, what's the difference? > > How can large cardinals be defined in terms of V (the Universe) when V > isn't a set? How are they cardinals if they aren't of sets? > > How can forcing's model be at once model and embedded in the model? > Wouldn't it then be irregular? > > Are there any results in recursion/computation that can't simply (or > not so simply) be framed in asymptotics without appeal to transfinite > cardinals? > > Are there any results not of transfinite cardinals, solely due > transfinite cardinals? >
Looking to http://cantorsattic.info , this is a nice resource for looking to the day's definitions of what would be "large cardinals", were they sets or cardinals.
A particular notion of the "large cardinals" is that they are defined in terms of embeddings into the set-theoretic universe (and correspondingly in set theory with classes the set and class universe, except where the distinction or existence is ignored). Then, the Kunen inconsistency that tops Cantor's attic is that there are no non- trivial embeddings from the universe V to itself. Yet, identity as trivial, sees then any pair-wise switch of elements from identity, eg for a and b that a->b and b->a, being as well an embedding. Then, if those are all trivial, then so is any function f: S -> S from set S to itself, which is a rather overbroad definition of "trivial". Would that remain elementary, in that the function preserves model isomoprhism, it would where the theorems of the resulting structure so modeled were the same. Then basically for groups or other features establishing isomorphisms for all relevant theorems of the structure, the pair-wise switch among elements identical under isomorphism, would yield non-trivial elementary embeddings. Basically that is as to whether, for example, the elements of Z_2, can be unique as elements of Z_2, from a sub-theory of ZF defining only the binary: that replacing all the 1's with 0's and 0's with 1's is structurally indistinguishable, under that all the coded results have the same import.
f: x e V -> V: x -> x (trivial identity) f: x e V -> V: x -> ~x (trivial? opposition)
Then, if each structure eg Z_2 various under isomorphism isn't a leaf or totally uniquely typed, then all mathematical structures are of a single unified structure, then that structure as consistent as structure itself could be: would be complete. As there exists ismorphisms thus elementary embeddings in those structures in vacuo, there exist all their regular 1-1 compositions defining elementary embeddings, in their concreteness.
The Universe would be irregular and be its own powerset. This is known as Cantor's paradox, that there can't be a universe as a regular/ well-founded set because uncountality depends on well-foundedness (and Regularity and Infinity are the only axioms of ZF restricting comprehension of quantification). Now, the Universe exists, simple and structurally from that anything exists. So, the universe is not a well-founded set, though defined by all its elements, it's naively a set. Large cardinals aren't set nor cardinals, of ZF.
Basically then from the upper attic of Kunen inconsistency to Con(ZFC): "Every model of ZFC contains a model of ZFC as an element", that models of ZFC are not standard nor well-founded, that's just a re- phrasal of Russell's paradox, that the collection of all the well- founded sets (which ZFC is) would contain itself: as alluded to there it does, then there's the simple question: why doesn't that imply ~Con(ZFC)?
I suggest you review the notion of forcing, in logic, and as to whether structurally, that breaks things. The simplest model: is a working, structural model. The Universe: is, what it is.