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JT
Posts:
1,388
Registered:
4/7/12


Re: Few questions on forcing, large cardinals
Posted:
Mar 19, 2013 11:37 AM


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 illfounded, 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/wellfounded 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.



