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Topic: Notation
Replies: 14   Last Post: May 27, 2013 1:46 AM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Notation
Posted: May 25, 2013 9:36 PM
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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 eta-th 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
> 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 non-empty class *C* has a member-minimal
> 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:
> 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
> definition-in-use 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

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