[FOM] 555: Foundational Methodology 6/Maximality

[Harvey Friedman]

I now want to jump way forward and plunge into the thicket of
"intrinsic maximality of the set theoretic universe".

As I continue to say, this notion is rather problematic, and has many
seductive formulations that are in fact inconsistent.

There is a simple formulation that makes really good sense in a
focused situation involving a tiny - but very important - part of the
set theoretic universe.

This is the restricted world of subsets of N = nonnegative integers.

Looking at our world POW(N) of subsets of N, we can arguably IMAGINE
that they have all been enumerated in a sequence, x_0, x_1, ...,
forming {2^i x 3^j: i is in x_j}. We call this subset of N, the join
of x_0,x_1,... .

Normally, we think of the join {2^i x 3^j: i is in x_j}, as not being
among x_0,x_1,... . However, this does require an hypothesis on
x_0,x_1,... .

THEOREM 1. There exists a Boolean algebra {x_0,x_1,...} containing any
given countable subset of POW(N), whose join is x_0.

THEOREM 2. If {x_0,x_1,...} satisfies RCA then it does not contain its join.

The kind of "maximality" here that seems interesting says, informally, that

*any imaginary extension {y_0,y_1,...} of {x_0,x_1,...} is similar to
{x_0,x_1,...}*

This still leaves a lot of leeway in precise formulations, and so we
need to go into a possibly lengthy exploratory phase with this.

For such a detailed exploratory phase, it seems imperative that we
have focused on such a modest situation - POW(N). But already here,
there is a lot of substance.

We have found the most traction in a kind of "anti-maximal idea*! This
is quite striking - in trying to get traction out of "maximality" we
come up with another kind of informal idea, that can be argued to be
somewhat related, but prima facie goes in the opposite direction in
some sense!!

Such is the nature of exploratory foundations.

**our universe POW(N) has an imaginary extension which is very much
like our universe POW(N)**

In fact, a lot of the substance being uncovered here is very close to
work I did in the 1980's concerning necessary uses of large cardinals
in the Borel world. There in order to show incompleteness, I
considered as a tool, various kinds of towers of countable subsets of
POW(N). Not quite the same as the ones that we are going to discuss
here, but with some similarities.

DEFINITION. Let A,B be countable subsets of POW(N). We say that B is a
rich extension of A if and only if
i. A,B satisfy ACA.
ii. A is an elementary substructure of B.
iii. B contains the join of some enumeration of A.

Of course, in i,ii, we use arithmetic on N.

There is also a parallel development that is based on dropping ii, but
we don't take this up here.

THEOREM 3. Some A has a rich extension B. This statement is equivalent
to the existence of an omega model of Z_2.

We now push ** further. We demand that the imaginary rich extension B
of our A = POW(N)  be even more similar to A than merely being rich.

Note that B "sees" A (through some enumeration of A), and so in some
sense, B "sees" (A,B). Since B is "like A", A should also "see" some C
such that (C,A) is similar to (A,B).

Cleaning this up, we arrive at

PROPOSITION 4. There exists C,A,B such that
i. C containedin A containedin B are countable subsets of POW(N) satisfying ACA.
ii. There is an enumeration of C in A, and an enumeration of A in B.
iii. Every true statement in (C,A) about elements of C is true in
(A,B) about those same elements of C.

We can streamline this further as follows.

PROPOSITION 5. There exists countable subsets A,B,C of POW(N) such that
i. A,B satisfy ACA.
ii. There is an enumeration of A in B.
iii. Every true statement in (A,B) about elements of A is true in
(B,C) about those same elements of A.

THEOREM 6. Propositions 4,5 are between totally indescribable
cardinals and subtle cardinals in strength.

This can be iterated in the obvious way, with iterated philosophy, to
go through the SRP hierarchy (same as the subtle or ineffable cardinal
hierarchy).

This whole development is very closely related to the Conservative
Growth development, but just involving countable subsets of POW(N).
See
86. Conservative Growth: A Unified Approach to Logical Strength,
October 12, 2014, 15 pages.
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/

There we also climbed up through measurables and strong measurables,
and even much higher. A parallel development can be carried here just
in POW(N).

Already in Propositions 4,5, we see that in A,B,C individually, we get
"there are only countably many reals constructible in any given real".

But where does this leave the philosophy of maximality? Well, I just
don't know. For the meantime, it got replaced by some other philosophy
of a prima facie conflicting kind. And how coherent can this other
philosophy really be made? Well that remains to be explored.

************************************************************
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 555th 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 #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM

Harvey Friedman