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

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Posts: 1,968
Registered: 12/4/12
Re: Notation
Posted: May 25, 2013 2:22 PM
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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

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

H_a subset H_(a+1) subset P(H_a)

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

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

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

(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....


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