FOM: 156:Societies

[Harvey Friedman]

The way out of Russell's paradox discussed in posting #155 is sufficiently
simple that I considered it worthwhile to begin a study of interpretations
of set theory (with large cardinals) in ordinary thought. My first versions
had some awkwardness, and have been modified to make more common sense. Of
course, as the versions are adjusted, the interpretations need to be
modified, with corresponding modification in the underlying technical work.

The basic interpretation is:

A set is a person. The universe of sets is a society. x epsilon y is
interpreted as "the person x likes the person y".

In the versions in this posting, it is easily seen that the societies are
forced to have infinitely many people. In the next posting, #157, we work
exclusively with socieities that have only finitely many people.

1. LIKES AND WEALTH IN SOCIETY.

Below we talk of wealth as one basis of comparison between people. However,
one could also use various other comparisons, many of which are of an
ethical or religious nature.

TERSE VERSION

In this society, one person likes or doesn't like another, and one person
is richer or poorer or of the same level of wealth as another.

For any given group, there are people of every level of wealth who, among
the persons richer than themselves, like the members and don't like the
nonmembers.

In any given group with arbitrarily poor members, there exists two
arbitrarily poor members, where every member liked by the richer of the two
is liked by the poorer of the two.

[This is to be interpreted either as a statement in set theory (society is
a set, and group is any subset of society), or as a scheme in predicate
calculus without equality (society is the domain, and group is any
definable subset of society).]

As a statement in set theory, this is equivalent, over a suitable weak
fragment of ZFC, to the existence of a subtle cardinal. And the least
cardinality of such a society is the least subtle cardinal.

As a scheme, it is mutually interpretable with ZFC + the scheme of
subtlety. This system is slightly weaker than ZFC + "there exists a subtle
cardinal".

EXPANDED VERSION (meaning unchanged).

In this society (of people), one person (in the society) likes or doesn't
like another (person in the society), and one person (in the society) is
richer or poorer or of the same level of wealth as another (person in the
society).

For any given group (of people in the society), there are people of every
level of wealth (i.e., of every level of exact wealth) who, among the
personhs richer than themselves, like the members (of the given group) and
don't like the nonmembers (of the given group).

In any given group (of people in the society) with arbitrarily poor members
(i.e., there are members of the group at least as poor as any given
person), there are two arbitrarily poor members (i.e., two members of the
group at least as poor as given person), where every member (of the given
group) liked by the richer of the two is liked by the poorer of the two.

2. SOCIETY WITH REPRESENTATIVES.

In this society, a subset of the people called the representatives, has
been chosen.

The "groups" are logically defined in terms of the like relation, wealth
comparison. Specific people can be mentioned when defining any group. We
will be interested in which specific people are mentioned when defining a
given group.

Nobody is poorer than all representatives.

There is a person who is not of the same wealth as any representative.

For every given person, somebody likes the representatives liked by the
given person and nobody else.

For any given person, there is a representative who likes the
representatives liked by the given person and doesn't like the
representatives not liked by the given person.

Every nonempty group that is defined using only representatives has a
member that is a representative.

For any given group there are people  of any given level of wealth who,
among the persons richer than themselves, like the members and don't like
the nonmembers.

[This is to be interpreted as a scheme in predicate calculus without
equality. It can also be interpreted as a statement in set theory, but
where group has the specific meaning given in the story; i.e., not as
arbitrary subset. As a scheme in predicate calculus with equality, this
interprets ZF + "there is an elementary embedding of a rank into a higher
rank with a critical point", and is interpretable in ZFC + "there is a
nontrivial elementary embedding of a limit rank into a higher limit rank".
This is easily seen to be stronger than ZFC + "there exists arbitrarily
large measurable cardinals". Using work of Woodin in large cardinal theory
without choice, it is stronger than ZFC + "there are arbitrarily large
Woodin cardinals". As a statement in set theory, if there is such a society
then there is a countably infinite one. Furthermore, as a statement in set
theory, it follows from the consistency of ZFC + "there is an elementary
embedding of a limit rank into a higher limit rank" and implies the
consistency of ZFC + "there exists arbitrarily large measurable cardinals".
Using work of Woodin in large cardinal theory wihtout choice, it implies
the consistency of ZF + "there is an elementary embedding of a rank into a
higher rank with a critical point".]

*********************************************

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This is the 156th in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones counting from #100 are:

100:Boolean Relation Theory IV corrected  3/21/01  11:29AM
101:Turing Degrees/1  4/2/01  3:32AM
102: Turing Degrees/2  4/8/01  5:20PM
103:Hilbert's Program for Consistency Proofs/1 4/11/01  11:10AM
104:Turing Degrees/3   4/12/01  3:19PM
105:Turing Degrees/4   4/26/01  7:44PM
106.Degenerative Cloning  5/4/01  10:57AM
107:Automated Proof Checking  5/25/01  4:32AM
108:Finite Boolean Relation Theory   9/18/01  12:20PM
109:Natural Nonrecursive Sets  9/26/01  4:41PM
110:Communicating Minds I  12/19/01  1:27PM
111:Communicating Minds II  12/22/01  8:28AM
112:Communicating MInds III   12/23/01  8:11PM
113:Coloring Integers  12/31/01  12:42PM
114:Borel Functions on HC  1/1/02  1:38PM
115:Aspects of Coloring Integers  1/3/02  10:02PM
116:Communicating Minds IV  1/4/02  2:02AM
117:Discrepancy Theory   1/6/02  12:53AM
118:Discrepancy Theory/2   1/20/02  1:31PM
119:Discrepancy Theory/3  1/22.02  5:27PM
120:Discrepancy Theory/4  1/26/02  1:33PM
121:Discrepancy Theory/4-revised  1/31/02  11:34AM
122:Communicating Minds IV-revised  1/31/02  2:48PM
123:Divisibility  2/2/02  10:57PM
124:Disjoint Unions  2/18/02  7:51AM
125:Disjoint Unions/First Classifications  3/1/02  6:19AM
126:Correction  3/9/02  2:10AM
127:Combinatorial conditions/BRT  3/11/02  3:34AM
128:Finite BRT/Collapsing Triples  3/11/02  3:34AM
129:Finite BRT/Improvements  3/20/02  12:48AM
130:Finite BRT/More  3/21/02  4:32AM
131:Finite BRT/More/Correction  3/21/02  5:39PM
132: Finite BRT/cleaner  3/25/02  12:08AM
133:BRT/polynomials/affine maps  3/25/02  12:08AM
134:BRT/summation/polynomials  3/26/02  7:26PM
135:BRT/A Delta fA/A U. fA  3/27/02  5:45PM
136:BRT/A Delta fA/A U. fA/nicer  3/28/02  1:47AM
137:BRT/A Delta fA/A U. fA/beautification  3/28/02  4:30PM
138:BRT/A Delta fA/A U. fA/more beautification  3/28/02  5:35PM
139:BRT/A Delta fA/A U. fA/better  3/28/02  10:07PM
140:BRT/A Delta fA/A U. fA/yet better  3/29/02  10:12PM
141:BRT/A Delta fA/A U. fA/grammatical improvement  3/29/02  10:43PM
142:BRT/A Delta fA/A U. fA/progress  3/30/02  8:47PM
143:BRT/A Delta fA/A U. fA/major overhaul  5/2/02  2:22PM
144: BRT/A Delta fA/A U. fA/finesse  4/3/02  4:29AM
145:BRT/A U. B U. TB/simplification/new chapter  4/4/02  4:01AM
146:Large large cardinals  4/18/02  4:30AM
147:Another Way  7:21AM  4/22/02
148:Finite forms by relativization  2:55AM  5/15/02
149:Bad Typo  1:59PM  5/15/02
150:Finite obstruction/statisics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02
152:sin  10:35PM  6/8/02
153:Large cardinals as general algebra  1:21PM  6/17/02
154:Orderings on theories  5:28AM  6/25/02
155:A way out  8/13/02