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Topic: some amateurish opinions on CH
Replies: 57   Last Post: Apr 16, 2013 8:12 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: some amateurish opinions on CH
Posted: Apr 11, 2013 3:22 PM

On 4/11/2013 10:19 AM, Dan wrote:
> I hold that ideally , when we don't know of the necessity of a (that
> is , not (?a ) holds) , that we should know that it is necessarily so
> that we don't know of the necessity of a , that is , (? not (?a )) .
> This makes "classical logic + ?" non-monotonic .

Pretty good.

http://en.wikipedia.org/wiki/Non-monotonic_logic

The pragmatic philosophers, Pierce and James, had issues
with representation of classical logic precisely because
of some of the types of non-monotonicity in the wikipedia
article.

> It also doesn't
> prevent anyone from exploring various consequences of either CH or
> (not CH) . My assumption was it is possible to formulate some
> questions such that, due some inherent circularity , are rendered
> unanswerable . Now, this leads to two possible situations : either
> we know of a statement's truth or falsity, or we know that we can't
> know of a statement's truth or falsity (as is the case with
> superdeterminism) . It would be sadistic to assume any other
> possibility (such as "we can't know , and we can't know that we can't
> know).
>

That makes sense.

I am afraid GCH may be precisely one of those
statments rendered unanswerable by the inherent
circularity issue.

Goedel's 1939 assumes denotations.

First-order logic assumes denotations.

So, the assumptions occur inside and outside of
models.

Intuitionistic arithmetization methods are invoked
to eliminate any defined constant symbols of the
notion of set theory.

But, the entire sense of model theory is that
truth derives from compositionality of terms
and compositionality of formulas in relation to
an assignment of extra-logical names. The
manipulation of eliminating descriptive definitions
defeats the model theory somewhat.

But, if one considers the extra-logical names,
then one is faced with a consistent global labelling
phenomenon in the sense of names as unique identifiers.
Given names, permutations abound. But, what is assumed
is a canonical sequential order corresponding to the
assignment of names.

In a sense, then, no model of set theory can have
greater plenitude than its ordinals. Relative to
definability within the model, this yields V=OD
(ordinal definability).

Kunen points out that V=OD requires one to know all
the sets of V. So, it is second order.

Relative to predicative instantiation in second
order, L=HOD (constructible universe = hereditarily
ordinal definable).

GCH is true in L. But, if my assessment is reasonable,
it is because of the issue of denotation. Note that
this is also related to a property of a global
canonical ordering. GCH has a close affinity with
the axiom of choice.

http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry#Relation_to_the_.28Generalised.29_Continuum_Hypothesis

So, the presupposition on extra-logical naming
is a strong assumption.

There is a theorem investigating the relation of the
axiom of choice to intuitionistic set theory via category
theory.

The axiom of choice can introduce classical complementation,
and, therefore, classical logic into the system

http://en.wikipedia.org/wiki/Diaconescu-Goodman-Myhill_theorem

http://www.ams.org/journals/proc/1975-051-01/S0002-9939-1975-0373893-X/S0002-9939-1975-0373893-X.pdf

So, these questions I have had with respect to denotation
appear to be related to the classical logic itself. It
may even be that "assuming choice" or "assuming choice via GCH"
is equivalent to assuming CON(T) with respect to a theory.

The sense of this analysis is that GCH is a question so
bound up with model theory that its independence
corresponds with your statement of circularity.

I have no problem with circularity, however. It only needs
to be introduced relative to an understanding of its
epistemic role.

http://en.wikipedia.org/wiki/M%C3%BCnchhausen_Trilemma

Date Subject Author
4/7/13 fom
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 Bergholt Stuttley Johnson
4/7/13 dan.ms.chaos@gmail.com
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 dan.ms.chaos@gmail.com
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 dan.ms.chaos@gmail.com
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 Virgil
4/8/13 dan.ms.chaos@gmail.com
4/8/13 mueckenh@rz.fh-augsburg.de
4/8/13 dan.ms.chaos@gmail.com
4/8/13 mueckenh@rz.fh-augsburg.de
4/8/13 dan.ms.chaos@gmail.com
4/8/13 mueckenh@rz.fh-augsburg.de
4/8/13 Virgil
4/8/13 Virgil
4/9/13 apoorv
4/8/13 Virgil
4/7/13 Virgil
4/9/13 Guest
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/10/13 Guest
4/10/13 dan.ms.chaos@gmail.com
4/10/13 fom
4/10/13 JT
4/11/13 apoorv
4/11/13 dan.ms.chaos@gmail.com
4/11/13 apoorv
4/11/13 fom
4/15/13 apoorv
4/15/13 fom
4/16/13 Shmuel (Seymour J.) Metz
4/16/13 fom
4/7/13 Virgil
4/7/13 William Elliot
4/7/13 fom
4/7/13 fom
4/8/13 William Elliot
4/8/13 fom
4/9/13 William Elliot
4/9/13 fom
4/9/13 William Elliot
4/9/13 fom
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/10/13 fom
4/11/13 dan.ms.chaos@gmail.com
4/11/13 fom
4/11/13 dan.ms.chaos@gmail.com
4/11/13 fom
4/9/13 fom