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Re: Few questions on forcing, large cardinals
Posted:
Mar 23, 2013 5:34 PM


On Mar 19, 8:37 am, JT <jonas.thornv...@gmail.com> wrote: > 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.
In a sense, infinity _is_ the numbers. Start from even more fundamental objects than natural numbers as elements. Like the numbers, they are as different as they can be and as same as they can be, where they are each different in not being any other and each same in being defined by that difference. There's no stop to that, it's gone on, forever. Then, in a way like when you look into the void, it looks into you, for there to be numbers counting upward, to infinity, there are numbers counting backward, from infinity. In a very fundamental sense, then infinity is _in_ the numbers.
As well, number theorists, who work on theorems about the structure of natural integers, sometimes define a point at infinity, on the line of numbers. The line of numbers goes on forever and contains only numbers, the point at infinity is on it. Sometimes, for example, the theorems are about there being a prime number at infinity, others, a composite and even the composite of all the numbers.
In real analysis, there are as well considerations of there being the point or points at infinity and negative infinity. This is even with staying in what otherwise is a Euclidean geometry with the parallel postulate, where other geometries find there isn't even the parallel postulate that parallel lines don't meet (at some point at infinity).
For proofs of induction and for bounded cases, the remarkable thing is that results are derived with only needing finite and unbounded elements. But, there are true results of these objects, that may well best be explained, with infinity, in the numbers.
Regards,
Ross Finlayson



