On 5/3/2013 5:03 PM, Graham Cooper wrote: > On May 3, 8:15 pm, fom <fomJ...@nyms.net> wrote: >> On 5/3/2013 2:43 AM, Graham Cooper wrote: >> >> >> >>> Its not possible to test Equality by Extension in the inf. case. >> >> That is correct Herc. >> >> On the other hand, it is not possible to interpret >> the universal quantifier as a universal statement if >> it is interpreted as a course-of-values. >> >> Aristotle wrote this. It is ignored by a certain >> contingent of the mathematical community who merely >> argues on the basis of beliefs concerning infinity. >> >> You know well that any computer system balances >> choices that affect performance. Relational databases >> run faster on logic chips optimized for integral >> arithmetic as opposed to floating point. The analogy >> applies here. >> >> Brouwer had been clear concerning how the effectiveness >> of working with finite sets differed from working >> with infinite sets. But, the reason infinity enters >> mathematics is because it is how the identity relation >> is extended to convey the geometric completeness of a >> line when used to represent the real number system. >> >> Infinity does not arise because of testability. It >> arises because of the nature of the identity relation. > > > If there are more SETS in ZFC than FORMULA in ZFC > (David C Ullrich) > > ZFC FORMULA | ZFC SETS > > 1 ___________ a i > 2 ___________ b p q r > 3 ___________ c j n > 4 ___________ d s t k u v > 5 ___________ e z w > ... > > THEN WHAT DO YOU MEAN BY ... > > A SET OF ZFC ? >
I actually agree with you somewhat here.
Nevertheless, if one restricts to countable models, then it is clear that there must be real numbers not represented. In particular, it is consistent that there will be finite sequences that are represented and that form what is called a generic set. These sets have a convergence property that permits the construction of forcing models. Cohen used this notion to derive generic reals.
So, with the current axiomatization of ZFC, it is usually possible to take advantage of countable ground models in such a way that one can form a forcing model that includes all of the original sets of the ground model and at least one set not in the ground model.
This is one reason I do not agree with the received paradigm concerning the standard account of identity,
you will find that Frege and Russell did not view existence as a first-order property. So, one finds one's self in the awkward trade-off between a comfortable first-order paradigm for logic and a second-order paradigm to establish a fixed notion of identity.
Hence, formulas alone are not enough (in my non-standard views of these things).