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Re: Notation
Posted:
May 25, 2013 2:22 PM


On 5/24/2013 10:57 PM, William Elliot wrote: > V is the FC 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.
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



