Date: May 25, 2013 9:36 PM Author: William Elliot Subject: Re: Notation 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

>

> http://plato.stanford.edu/entries/identity-relative/#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 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:

>

> 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

> 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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