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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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Posts: 2,720
Registered: 2/15/09
Re: Few questions on forcing, large cardinals
Posted: Mar 19, 2013 11:17 AM
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On Mar 19, 1:12 am, fom <> wrote:
> On 3/19/2013 2:18 AM, Ross A. Finlayson wrote:

> >
> > "Rather, the focus is on the connections between the combinatorial
> > properties of the partial order of conditions and structural
> > properties of the extension."

> > "With forcing so expanded into the interstices of set theory and the
> > method so extensively amended from the beginning, what is the "it" of
> > Cohen's forcing and his individual achievement? Cohen discovered a
> > concrete and widely applicable means of operationally extending a
> > standard model of set theory to another without altering the ordinals.
> > The central technical innovation was the definable forcing relation,
> > through which satisfaction for the extension could be approached in
> > the ground model. Cohen's achievement was thus to be able to secure
> > properties of new sets without having all of their members in hand and
> > more broadly, to separate and then interweave truth and existence."

> On page 360 he speaks of having to
> think of truth in new way.
> One result of forcing has been to
> consider "truth persistence under
> forcing".  To the best of my knowledge,
> this typifies some of Woodin's work.
> "Partiality, Truth, and Persistence" by Langholm
> Some information about "names" and "descriptions" in
> relation to presupposition and the strong Kleene truth
> definition used by Langholm
> And yes, forcing is unobjectionable when you redefine truth.
> But, no one told anyone.

If you might elucidate on that, it may help to establish the context a
bit more firmly to the gallery.

A model of ZF might be ill-founded, or even where it's not, it
basically includes ZF and all its sets which is the Russell set. (As
it is, upon inspection.) Cohen's forcing (of an ordinal structure)
would have that then the resulting items are modeled as ordinals.

Then, of large cardinals, it is somewhat a misnomer, cardinal, when
they're not of regular/well-founded objects of the set theory, yet
forced to be.

As an extension of Skolemization, Cohen's forcing is upward (past the
entire model of regular set theory), yet, there is also downward in L-
S. Why doesn't that correspond to the irregular and "Kunen
inconsistency" (Cantor paradox) forced into the regular? This may be
considered with regards to Levy collapse.

For infinity in the numbers, we start counting, and it doesn't end,
from that there is infinity, in the numbers.

Are there, any results solely due transfinite cardinals, not of
transfinite cardinals? Via forcing, there are results of transfinite
cardinals, not due transfinite cardinals, but only of, transfinite


Ross Finlayson

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