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Topic: Few questions on forcing, large cardinals
Replies: 17   Last Post: Mar 30, 2013 1:21 PM

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ross.finlayson@gmail.com

Posts: 1,218
Registered: 2/15/09
Re: Few questions on forcing, large cardinals
Posted: Mar 30, 2013 1:21 PM
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On Mar 16, 3:44 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> On Mar 10, 5:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> wrote:
>
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> > 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 tohttp://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.
>


Then, the notion that the Kunen inconsistency establishes that there
are no non-trivial embeddings of elements, has these notions of
counterexamples: that there are finite or infinite isomorphic
structures, and that opposition isn't trivial.

Opposition is to map each element v of the universe V to the element V
\v. As it's a universal set, it contains as elements V\v, basically
the complement of elements {v}. Then, what about when v is itself an
irregular collection, eg of ordinals? The idea here is to figure out
where V\v = v. That gets into notions like: the Russell set, with
and without its sputnik of quantification, that R_realized \r =
R_specification, but that: r = R_specification, that R_realization \
r = R_specification = r, for R the Russell set and r itself. These
self-complementing sets (in the universe) are then a similar notion as
to the collection of groups with a natural operation of composition
aside auto-annihilation: Grp as a group. Basically in the universal,
there is that disjoint themselves, some objects are still themselves,
and that, in union, some non-empty objects union themselves are
empty. These are structural features that follow from deduction.

Then for isomorphic structures, that is as simple as any system with a
language with two elements, that any two elements satisfy the
properties of being distinct elements with regards to the structure,
that any are interchangeable. Unless the stucture is actually all the
possible structure that could be, which it would thus be, there is a
non-trivial (non-identity) simple transposition of those elements.
Models are concrete, or not, and if not, there are non-trivial
elementary embeddings in the universe of them, under isomorphism, and
if so, then universe as concrete model is irregular: and thus is the
domain.

That doesn't much speak to whether it's hypocritical to have large
cardinals, bereft properties of cardinals and sets, forced into the
space of regular cardinals (well-founded cardinals as it were). There
either are or aren't "inconsistent multiplicities": defined by their
elements, that the universe exists or the "domain principle" is true
has merit as a truism. To apply a class distinction, for example to
ordinals as defined by their elements and containing lesser ordinals,
sees that there is courtesy their construction: no difference, to
have quantification over elements, or not, and that it's inconsistent
to have both classes and sets without having both defined by their
elements and their structure.

Regards,

Ross Finlayson



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