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Re: Notation
Posted:
May 25, 2013 9:36 PM


On Sat, 25 May 2013, fom wrote:
> On 5/24/2013 10:57 PM, William Elliot wrote: > > V is the ZFC universe. > > L is the constructible universe. > > L_omega0 is the omega_0th level of the constructible universe. > > > > Correct or needing correcting? > > Your apparently simple question seems to have generated > some interesting replies. > Seemingly off the mark because I'm asking about notation and not about theory. When constructing the constructible universe we start with
L_0 = emptyset = 0 L_1 = { 0, {0} }. Let {0} = 1. L_2 = { 0, 1, {1}, {0,1} }
L_(n+1) = L_n \/ P(L_n) L_omega0 = \/_n L_n.
Is L_eta the correct notation for the constructible universe to the etath level?
> Taking ZFC according to standard logical axiomatizations > gives the universe as > > V = {x: x=x} > > in class comprehension form. You may observe that this > is merely a restatement of the reflexive law from the > standard account of identity > > http://plato.stanford.edu/entries/identityrelative/#1 > > which is invoked by both Jech and Kunen in their texts. > > Aatu is correct in referring to "the cumulative hierarchy". > > However, if you look at numbered formula 11.16 in Jech, > "cumulative hierarchy" can be generalized: > > "A transfinite sequence <H_a : a element Ord> is > called a cumulative hierarchy if > > (i) > H_a subset H_(a+1) subset P(H_a) > > (ii) > if a is a limit ordinal, then H_a = \/_(b<a) H_b" > > One should observe the role of the power set > operation in (i). Aatu's reference to a singular > hierarchy is the one given by the situation > where > > H_(a+1) = P(H_a) > > To say that V is the cumulative hierarchy involves > the axiom of regularity/foundation. Jech lists > 3 lemmas to establish that every (pure) set has > a rank in the cumulative hierarchy: > > Lemma 9.1 > For every set S there exists a transitive > set T such that (S subset T) > > Lemma 9.2 > Every nonempty class *C* has a memberminimal > element > > Lemma 9.3 > For every x there is an a such that (x element V_a): > > V= \/_(a element Ord) V_a > > > Peter made certain observations with regard to a > second comment from Aatu responding to Barb. Those > numbered observations may be expanded upon as > follows: > > (i) a theory of ordinals is presupposed, > > To this one may add a quote from Solovay's introduction > to Goedel's papers: > > "Referring to his work on the consistency of CH > [Goedel] says, 'However, as far as, in particular, > the continuum hypothesis is concerned, there was > a special obstacle which really made it practically > impossible for constructivists to discover my > consistency proof. It is the fact that the > ramified hierarchy, which had been invented > expressly for constructive purposes, has to be > used in an entirely nonconstructive way.' The > essentially nonconstructive element lies in the > use of arbitrary ordinals as the levels in > Goedel's extension of the ramified theory." > > > (ii) it is assumed that one knows what the power set of > a set is (not a problem for finite sets) which determines > how "fat" V is, > > I would personally disagree with Peter's parenthesized > remark. However, his description of "fatness" corresponds > with Jech's generalized notion of a cumulative hierarchy > in which the power set operation is the upper bound in > the ordering > > H_a subset H_(a+1) subset P(H_a) > > The power set operation is viewed as impredicatively > defined and, as such, is considered objectionable to > predicative philosophies. The predicative notion would > be that there is some framework that is applied to > *given* objects. > > The best I can do to explain this predicativist > position is to refer to "priority pluralism" in > the link: > > http://plato.stanford.edu/entries/monism/#PriorityMonism > > What tends to bother me about this view is that the > role of quantifiers in set theory is necessarily > different from the role of quantifiers in other > theories. Quine makes the distinction by referring to > "general quantification" as distinguished from quantification > in typed theories. Goedel's platonism is clearly > "general quantification". > > > (iii) it says nothing about how "long" the ordinals are over which the final > union is taken, > > Actually, this is not quite correct. Aatu probably knows more than > I with respect to Kunen's inconsistency result. There can be no > elementary embedding of the universe into itself. > > I may have stated that incorrectly. > > Peter's remark, however, applies in the sense that transfinite > arithmetic is formulated in the tradition of "number system" as > it became pluralized in the 19th century with arithmetical systems > such as complex numbers and quaternions. > > Kant summarizes the problem nicely: in relation to a singular > judgement, a universal judgement is infinite. Extending that > sense to completed infinities (which Kant denied) simply > exchanges "absolutely infinite" with "infinite". > > > (iv) what reason is there to suppose that _all_ sets are in there? > > This is a matter of philosophy on the broadest level. > > The criticism of "definiteness" against Zermelo's original > formulations had been met by restricting the sense according > to which a set may be referenced (Skolem, among others). > Presumably, comprehension schemes such as > > {x: phi(x)} > > are restricted to formal languages. In this context, the > meaning of "set" in "set theory" is no more than the > definitioninuse given by the assumptions of the theory. > Then, Jech's lemmas quoted above prove that every set is > in the cumulative hierarchy. > > Then, however, model theory raises its ugly head and the > question of how to interpret a universal quantifier becomes > problematic. Technically (along the lines of Barb's notions) > "classes" are mere grammatical forms. They are simply formulas > and not objects of the theory. Relative to any "set model" > of the axioms, one can distinguish sets and classes on the > basis of "subsets of the model which are elements of other subsets > of the model" and "subsets of the model which are not elements > of other subsets of the model". > > Such a simple question.... > > chuckle > > > > > > > > >



