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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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JT

Posts: 1,041
Registered: 4/7/12
Re: Few questions on forcing, large cardinals
Posted: Mar 19, 2013 11:37 AM
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On 19 mar, 16:17, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> On Mar 19, 1:12 am, fom <fomJ...@nyms.net> wrote:
>
>
>
>
>
>
>
>
>

> > On 3/19/2013 2:18 AM, Ross A. Finlayson wrote:
>
> > >http://math.bu.edu/people/aki/14.pdf
>
> > > "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
>
> >http://books.google.com/books?id=AOGpRxZyfFYC&pg=PA43&lpg=PA43&dq=tru...
>
> > Some information about "names" and "descriptions" in
> > relation to presupposition and the strong Kleene truth
> > definition used by Langholm

>
> >http://plato.stanford.edu/entries/presupposition/#LocConDynTur
>
> > 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
> cardinals.
>
> Regards,
>
> Ross Finlayson


No the infinity is not ***in the numbers*** there is infinity but it
ain't numerical.




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