904: Foundations of Large Cardinals/2

[Harvey Friedman]

1. Conceptual Remarks concerning - here
2. Second Thoughts about Elementary Extensions versus Elementary
Equivalence - here
3. Two Universe Elementary Extension - here
4. Two Extended Universe Restricted Elementary Extension - #3
5. Double Completion Elementary Extension - foundations of ZF - #4
6. Multiple Universes/Completions Restricted Elementary Extension -#5
7. Two Universe Elementary Extension/complex - #6

1. CONCEPTUAL REMARKS CONCERNING 903

In 903 I didn't really discussed the general conceptual framework or
frameworks that led to that particular systematic investigation. I
have too many pressing matters right now to really get into the
details of the reversals suggested there.

The conceptual picture that prompted 903 is simply this. Let's say we
start off with a or the set theoretic universe V. There is the idea of
going further which has proved rather compelling and useful in the
rather limited sense of bringing in classes. By Russell we know that
this is really an enlargement, with the proper classes. Sometimes
used, but less often by far, are the classes of classes, and once
again, this is another enlargement.

Perhaps the right way to look at this is that we can reflect on the
concept of V, the set theoretic universe, and instead of viewing it as
the END, we can instead view it as the BEGINNING, or RESTART. So we
can rename V as V_0, and reconceive of the set theoretic universe
again, yielding V_1.

In this discussion, we are taking for granted a notion of set
theoretic universe that comes with the ZFC axioms (or ZF axioms) as
given, with the conceptual picture of V(theta) for some strongly
inaccessible cardinal theta, in mind, in terms of a proper class
associated with that set theoretic universe.

So according to this enlargement or explosion picture, where V_0
explodes to V_1, the second set theoretic universe, not only is every
V_0 element a V_1 element, and the epsilon relation preserved, but
also every V_1 element of a V_0 element is already a V_0 element. Thus
we have the picture of V_1 as an end extension of V_0.

This picture of exploding end extensions is extremely familiar from
the most basic of elementary mathematics. Namely, we can start with
[0,1). This explodes to the end extension [0,2), and then we have the
unending series [0,1), [0,2), [0,3), ... . We can be doing this either
in the real numbers or in the rationals. And it is natural to attempt
to carry this out in the nonnegative integers. Here this presumably
leads to the foundations of a philosophical >> between nonnegative
integers.

So with just a finite series of strongly inaccessible V(theta) and
elementary equivalence, we would just get about V(kappa) where kappa
is the (beth_omega)+ strongly inaccessible (where the universes are V
of strongly inaccessibles).

But

V_0, V_1, V_2, ...

with just this elementary equivalence of infinite subsets (of these
V's) is quite strong. You get far more than iterated #'s as in 0# of
Jack Silver. With known inner model theory below a measurable
cardinal, presumably you get about the level of kappa arrow omega in
the partition calculus.

Actually we can just use the elementary equivalence of

V_0, V_1, V_2, ...
V_1, V_2, V_3,  ...

which has a particularly compelling philosophical meaning. Namely it
says that it doesn't make any difference whether we start with V_0 or
start with V_1. We cannot distinguished between these two.

Physically, this is like looking out from here or looking out from the
first epochal jump from here. We see exactly the same things.

The reversal ideas here go back to Donald Martin when he was reversing
just beyond analytic determinacy. The proof for all infinite subsets
uses Prikry forcing through measurables. Certainly using just co
finite infinite subsets we can get away with kappa arrows omega and
not use Prikry sequences.

Nothing much seems to be added here if we use elementary extensions
rather than just elementary eqivalence.

However if we use elementary extensions in the finite case then we
really do get something interesting. We can use

V_0, V_1, ..., V_n
V_1, V_2, ..., V_n+1

and get the SRP hierarchy of large cardinals this way.

To get to the level of measurable cardinal exactly, we view

V_0, V_1, V_2, ...
V_1, V_2, V_3,  ...

as elementary equivalent as a scheme over a master universe V in which
this infinite series is residing.

Alternatively, if we just stay locally in these series, we can use a
series of length omega + 1,, and use the elementary equivalence of

V_0, V_1, V_2, ..., V_omaga
V_1, V_2, V_3,  ..., V_omega

which are of length omega+1.

And as discussed in #903, we can use transfinite series and with some
care (regular restrictions) we can get higher order kinds of
measurable cardinals.

But I think the MOST COMPELLLING TAKEAWAY is just to write down the
semiformal pair

V_0, V_1, V_2, ...
V_1, V_2, V_3,  ...

with elementary equivalence as a scheme in the outer universe, and say
that we are firmly into medium large cardinals. If we just want the
level of ZC + there exists a measurable cardinal then we don't need to
use the scheme, but rather elementary equivalence over V(lambda +
omega), where lambda is the union of the ordinals of the V_n's.

2. SECOND THOUGHTS ABOUT ELEMENTARY EXTENSIONS VERSUS ELEMENTARY EQUIVALENCE

In 903, I expressed a great preference for elementary equivalence over
elementary extensions in this kind of foundations of large cardinals.
I did say that the reversals conjectured there would be considerably
easier with elementary extensions rather than elementary equivalence.
And in fact in the finite case, instead of staying well within ZF, we
would be at the level of the SRP hierarchy of large cardinals.

In thinking about this again, I think I see less of a gap,
philosophically, between elementary extension and elementary
equivalence here. So obviously I think both should be vigorously
investigated. Look at section 3 below, the paragraph marked ***.

3. TWO UNIVERSE ELEMENTARY EXTENSION

Now that I have warmed up to elementary extensions and not just
elementary equivalence, I want to focus on the two universe case,
V_0,V_1. In this section, in accordance with 1 above, both universes
are taken to be strongly inaccessible ranks (i.e., V(theta), theta
strongly inaccessible). A weakening of strongly Mahlo arises here.

DEFINITION 3.1. A cardinal kappa is definably strongly Mahlo if and
only if every closed unbounded subset of kappa that is definable over
(V(kappa),epsilon) with parameters from V(kappa), has a strongly
inaccessible element. On is definably Mahlo if and only if every
closed unbounded class of ordinals that is definable over (V,epsilon)
with parameters from V, has a strongly inaccessible element. The
former is formulated as a single formula, but the latter is formulated
as a scheme. We use n-definably to mean definable using at most n
quantifiers.

THEOREM 3.1. (Known).  Let V_1 be a proper elementary extension of
V_0, where V_0,V_1 are strongly inaccessible ranks on kappa < lambda
respectively. .
i. In V_0,V_1, On is definably Mahlo.
ii. In V_1, kappa is definably Mahlo.
iii. Let n be given. In V_0,V_1, there exists arbitrarily large
cardinals that are n-definably strongly Mahlo.

Proof: Let V_0,V_1 be as given with V_0 = V(kappa), V_1 = V(lambda).
To see that On is definably Mahlo in V_0, let S be a closed unbounded
class of ordinals definable over V(kappa), in the sense of V(kappa).
Then S extends to one over V(lambda) by elementary extension, and
kappa itself is an element by the closedness. Since kappa is strongly
inaccessible in V(lambda), by elementary extension we have that
V(kappa) satisfies that S has a strongly inaccessible element. It now
follows by elementary extension that since On is definably Mahlo in
V_0, On is also definably Mahlo in V_1. This also establishes ii.

For iii, fix n. Suppose alpha < kappa and according to V_0, there are
no n-definably strongly Mahlo cardinals > alpha. Then by elementary
extension, there are no n-definably strongly Mahlo cardinals > alpha
according to V_1. This contradicts ii. So iii holds for V_0. Hence by
elementary extension, iii holds for V_1. QED

But can we get an ordinary strongly Mahlo cardinal in V_0,V_1 out of
the hypotheses in Theorem 3.1? No.

THEOREM 3.2. (Known). There exists V_0,V_1 with the hypotheses of
Theorem 3.1, strictly below the first strongly Mahlo cardinal. Hence
these V_0,V_1 do not satisfy that there exists a Mahlo cardinal.

Proof: Let mu be the least strongly Mahlo cardinal. Let S be the class
of all alpha such that V(alpha) is an elementary substructure of
V(mu). Since mu is strongly inaccessible, S is closed and unbounded.
Hence S has arbitrarily large strongly inaccessible elements, and in
particular two. These yield the desired V_0,V_1. QED

In fact, we can easily see from Theorem 3.1, 3.2 that we are not going
to get anything more by looking at successive elementary extensions of
strongly inaccessible ranks. Theorem 3.1 immediately extends, and also
Theorem 3.2 extends because of the last sentence in its proof.

So we have fallen short of getting strongly Mahlo cardinals. So how
are we going further?

In section 4, we use V(kappa + 1) and V(lambda + 1) instead of the
present V(kappa) and V(lambda). This allows us to go much further, way
beyond strongly Mahlo cardinals, into second order indescribable
cardinals. In section 6, we use three universes (and more), and this
allows us to go yet much further.

With regard to using three universes V_0,V_1,V_2. Here we use that
(V_1,V_2) is a restricted elementary extension of (V_0,V_1). Here we
use parameters from V_0, and we view these two pairs as two sorted
structures. This is the same as using single sorted V_1,V_2, but each
augmented with unary predicates V_0,V_1, respectively - but the
elementary extension notion would have to be qualified. If we view
these instead as two sorted, then this use of parameters from V_0 is
like elementary extension for sorted structures where a single sort is
where the parameters are chosen form. But note that in any case, we
are talking about a natural kind of restricted elementary extension.
I.e., the parameters used are restricted to V_0 even though in
(V_0,V_1), no matter how construed, the full range of points is V_1.

***These considerations led us in 903 to strongly prefer elementary
equivalence rather than (restricted) elementary extension. However,
this does have us miss out on the SRP hierarchy in this kind of
foundations of large cardinals. Later we will take up the multiple
case where we get the SRP hierarchy.

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My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 904th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-899 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

900: Ultra Convergence/2  10/3/21 12:35AM
901: Remarks on Reverse Mathematics/6  10/4/21 5:55AM
902: Mathematical L and OD/RM  10/7/21  5:13AM
903: Foundations of Large Cardinals/1

Harvey Friedman