FOM: 116:Communicating Minds IV

[Harvey Friedman]

COMMUNICATING MINDS IV
by
Harvey M. Friedman
January 4, 2002

This posting is based on two minds communicating about the concept of
extensional classes.

The basic idea is clean and close to the Russell paradox.

Mind I forms a virtual class as usual, {x: phi(x)}, where x ranges over I's
classes, the bound variables range over I's classes, and there are free
variables - parameters - standing for some of I's classes. As usual, this
may not form one of I's classes by the Russell paradox. I.e., phi(x) may be
"x notin x".

We want a sufficient condition for {x: phi(x)} to form a class of I. We
will take this in the sense that we want there to be a class y of I such
that for all classes x of I, x in y if and only if phi(x).

Axiom scheme 6c asserts that the following condition is sufficient. We
first reinterpret phi(x) by mind II, where the parameters are unchanged,
but the bound variables now range over II's classes, and x ranges over II's
classes. We write this interpretation by mind II as phi*(y). The sufficient
condition is that if phi*(y) then y is one of mind I's classes.

I.e., the sufficient condition is that we don't pick up new classes when
reinterpreting by mind II.

Axiom scheme 6c goes further and asserts that not only does {x: phi(x)}
form a class of I, but it also forms a domain for full comprehension for
mind I.

Axiom scheme 6d also provides a domain for full comprehension. This time
the hypothesis is the exact opposite: that phi*(y) holds of some y that is
among II's classes but not among I's classes. The conclusion is that such a
y can be found so that we can assert full comprehension for mind I.

With 6c we get to ZFC, and with both 6c and 6d we get to measureable
cardinals and beyond.

NOTE: There are some related ways of getting to ZFC and also beyond
measureables that are somewhat stronger. These use full comprehension for
classes, whereas 6c and 6d just use full comprehension for certain classes.
These are presented in sections 15 and 16. Finally, in sections 17 and 18
we give formulations which are single sorted and meant to be as simple as
possible. The ones in 17 and 18 are somewhat further away from the two
minds idea.

13. I's CLASSES, II's CLASSES, SPECIAL COMPREHENSION.

We use variables x1,x2,... over I's classes.

We use variables y1,y2,... over II's classes.

The atomic formulas are of the following forms:

i. u = v, where u,v are variables of either sort.
ii. u in v, where u,v are variables of either sort.

We close under atomic formulas under connectives and quantifiers in the
usual way. We call this language L6. We use the standard 2 sorted predicate
calculus with equality appropriate for L6.

6a. (therexists y1)(x1 = y1).

6b. y1 = y2 iff (forall y3)(y3 in y1 iff y3 in y2).

6c. (forall y1)(phi* implies (therexists x1)(x1 = y1)) implies (therexists
xk+1)(forall x1)(x1 in xk+1 iff (phi and psi)), where phi is a formula of
L6 with all variables among x1,x2,...,xk, phi* is the result of replacing
each bound occurrence of xi by yi and each free occurrence of x1 by y1, and
psi is a formula of L6 in which xk+1 is not free.

THEOREM 13.1. The system 6a-6c is mutually interpretable with ZFC.

14. I's CLASSES, II's CLASSES, ADDITIONAL SPECIAL COMPREHENSION.

We add an additional restricted comprehension axiom scheme.

6d. (therexists y1)(phi* and not(therexists x1)(x1 = y1)) implies
(therexists y1)(phi* and not(therexists x1)(x1 = y1) and (therexists
x1)(forall x2)(x2 in x1 iff (x2 in y1 and psi))), where phi is a formula of
L6 with all variables among x1,x2,..., phi* is the result of replacing each
bound occurrence of xi by yi and each free occurrence of x1 by y1, and psi
is a formula of L6 in which x1 is not free.

THEOREM 14.1. The system 6a-6d can be proved consistent in, and hence is
interpretable in, ZFC + "there exists a nontrivial elementary embedding
j:V(kappa + 1) into V(lambda + 1). The system 6a-6d proves the consistency
of, and hence interprets, ZFC + "there ZFC + there exists arbitrarily large
Woodin cardinals".

15. I's CLASSES, II's CLASSES, COMPREHENSION, COMMUNICATION.

6a. (therexists y1)(x1 = y1).

6b. y1 = y2 iff (forall y3)(y3 in y1 iff y3 in y2).

6e. (therexists y1)(forall x1)(x1 in y1).

6f. (therexists x1)(forall x2)(x2 in x1 iff (x2 in x3 and phi)), where phi
is a formula of L6 in which x1 is not free.

6g. (therexists x1)(forall y1)(y1 in x1 iff (y1 in x3 and phi)), where phi
is a formula of L6 in which x1 is not free.

6h. (therexists y1)(forall y2)(y2 in y1 iff (y2 in y3 and phi)), where phi
is a formula of L6 in which y1 is not free.

6i. phi iff phi*, where phi is a formula of L6 whose variables are among
x1,x2,..., and phi* is the result of replacing each bound occurrence of xi
by yi.

Note that 6g implies 6f in the presence of 6a. Also 6f implies 6h in the
presence of 6i.

THEOREM 15.1. The system 6a,6b,6e-6i and the system 6a,6b,6e,6f,6i are each
mutually interpretable with ZFC.

16. I's CLASSES, II's CLASSES, COMPREHENSION, COMMUNICATION, DOMINANCE.

We consider a dominance axiom scheme.

6j. (therexists y1)(phi and (forall x1)(x1 not= y1)) implies (therexists
y1)(phi and (forall x1)(x1 not= y1) and (therexists x1)(y1 in x1)), where
phi is a formula of L6 with all free variables among y1,x1,x2,..., and all
bound variables among y1,y2,... .

THEOREM 16.1. The system 6a,6b,6e-6j and the system 6a,6b,6e,6f,6i,6j have
the properties in Theorem 14.1.

17. SINGLE SORTED SYSTEM CORRESPONDING TO ZFC.

Here we use variables x1,x2,..., membership, and the constant symbol W.
Call this single sorted language L7.

7a. (forall x3)(x3 in x1 iff x3 in x2) implies (x1 in x3 iff x2 in x3).

7b. (therexists x1)(forall x2)(x2 in x1 iff (x2 in x3 and phi)), where phi
is a formula of L7 in which x1 is not free.

7c. (x1,...,xk in W and (therexists xk+1)(phi)) implies (therexists
xk+1)(phi and x1 in W), where phi is a formula of L7 not mentioning W, and
all free variables are among x1,...,xk+1.

This is the system we discussed in #90:Two Universes, 6/23/00  1:34PM.

THEOREM 17.1. The system 7a-7c and the system 7b-7c are mutually
interpretable with ZFC.

18. SINGLE SORTED SYSTEM CORRESPONDING TO LARGE CARDINALS.

We strengthen 7c as follows.

7d. (x1,...,xk in W and (therexists xk+1 notin W)(phi)) implies
((therexists xk+1 in W)(phi) and (therexists xk+1 notin W)(phi and
(therexists xk+2 in W)(xk+1 in xk+2))), where phi is a formula of L7 not
mentioning W, and all free variables are among x1,...,xk.

THEOREM 18.1. The system 7a,7b,7d has the properties in Theorem 14.1.

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This is the 115th 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