[FOM] 559: Foundational Methodology 7/Maximality

[Harvey Friedman]

1. GENERAL STRATEGY.
2. THE LANGUAGE L0.
3. STRONGER LANGUAGES.

1. GENERAL STRATEGY

Here we present a way of using the informal idea of "intrinsic
maximality of the set theoretic universe" to do two things:

1. Refute the continuum hypothesis (using PD and less).
2. Refute the existence of certain large cardinals (in ZFC).

Quite a tall order!

Since I am not that comfortable with "intrinsic maximality", I am
happy to view this for the time being as an additional reason to be
even less comfortable.

At least I will resist announcing that I have refuted both the
continuum hypothesis and existence of certain extensitvely studied
large cardinals!

INFORMAL HYPOTHESIS. Let phi(x,y,z) be a simple property of sets
x,y,z. Suppose ZFC + "for all infinite x, there exist infinitely many
distinct sets which are pairwise incomparable under phi(x,y,z)" is
consistent. Then for all infinite x, there exist infinitely many
distinct sets which are pairwise incomparable under phi(x,y,z).

Since we are going to be considering only very simple properties, we
allow for more flexibility.

INFORMAL HYPOTHESIS. Let 0 <= n,m <= omega. Let phi(x,y,z) be a simple
property of sets x,y,z. Suppose ZFC + "for all x with at least n
elements, there exist m distinct sets which are pairwise incomparable
under phi(x,y,z)" is consistent. Then for all x with at least n
elements, there exist at least m distinct sets which are pairwise
incomparable under phi(x,y,z).

We can view the above as reflecting the "intrinsic maximality of the
set theoretic universe".

We will see that this Informal Hypothesis leads to "refutations" of
both the continuum hypothesis and the existence of certain large
cardinals, even using very primitive phi in very primitive set
theoretic languages.

2. THE LANGUAGE L0

L0 has variables over sets, =,<=*,U. Here =,<=* are binary relation
symbols, and U is a unary function symbol. x <=* y is interpreted as
"there exists a function from x onto y". U is the usual union
operator, Ux being the set of all elements of elements of x.

MAX(L0,n,m). Let 0 <= n,m <= omega. Let phi(x,y,z) be the conjunction
of finitely many formulas of L0 in variables x,y,z. Suppose ZFC + "for
all x with at least n elements, there exist m distinct sets which are
pairwise incomparable under phi(x,y,z)" is consistent. Then for all x
with at least n elements, there exist at least m distinct sets which
are pairwise incomparable under phi(x,y,z).

THEOREM 2.1.ZFC + MAX(L0,infinity,infinity) proves that there is no
(omega+2)-extendible cardinal.

More generally, we have

THEOREM 2.2.  Let 2 < log(m)+1 < n <= omega.
i. ZFC + MAX(L0,n,m) proves that there is no (omega+2)-extendible
cardinal. Here log(omega) = omega.
ii. ZFC + PD + MAX(L0,n,m) proves that the GCH fails at all infinite
cardinals. In particular, it refutes the continuum hypothesis.
iii. ii with PD replaced by higher order measurable cardinals in the
sense of Mitchell.

We are morally certain that we can easily get a complete understanding
of the meaning of the sentences in quotes that arise in the
MAX(L0,n,m).

Write MAX(L0) for "for all 0 <= n,m <= omega, MAX(L0,n,m)". Using such
a complete understanding we should be able to establish that ZFC +
MAX(L0) is a "good theory". E.g., such things as

1. ZFC + PD + MAX(L0) is equiconsistent with ZFC + PD.
2. ZFC + PD + MAX(L0) is conservative over ZFC + PD for sentences of
second order arithmetic.
3. ZFC + PD + MAX(L0) + "there is a proper class of measurable
cardinals" is also conservative over ZFC + PD for sentences of second
order arithmetic.

We will revisit this development after we have gained that complete
understanding. Then we will go beyond finite conjunctions of atomic
formulas in L0.

The key technical ingredient in this development is the fact that

i. GCH fails at all infinite cardinals is incompatible with
(omega+2)-extendiable cardinals (Solovay).
2. GCH fails at all infinite cardinals is demonstrably consistent
using much weaker large cardinals, or using just PD (Foreman/Woodin).

************************************************************
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 558th 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
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14  10:01PM

Harvey Friedman