This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 1
A COMPLETE THEORY OF EVERYTHING:
satisfiability in the universal domain
Harvey M. Friedman
October 10, 1999
friedman@math.ohiostate.edu
http://www.math.ohiostate.edu/~friedman/ 1. GENERAL REMARKS
Let LPC(=) be the usual language of predicate calculus with
equality, and PC(=) be a standard system of axioms and rules
and inference for predicate calculus with equality.
The completeness theorem determines those sentences that have
models with certain domains. Here are five cases:
1.
2.
3.
4.
5. Nonempty sets.
N.
Infinite sets.
Fixed infinite set.
Extensions of unary predicates. Very few basic principles are needed for these results. 4
needs the axiom of choice.
Note that 5 does not have the accepted clarity of meaning
that 14 have. In fact, we will be considering two major
distinct interpretations of 5. Nevertheless, the completeness
theorem applies to 5, since its proof is so robust.
We call 5 the domain interpretation.
Here we take the view that LPC(=) is applicable to structures
whose domain is too large to be a set. This is not just a
matter of class theory versus set theory, although it can be
interpreted as such, and this interpretation is discussed
briefly at the end.
We apply LPC(=) to the largest domain of all  the domain W
consisting of absolutely everything.
In order to make sense of this, we need the concept of
arbitrary predicate of several variables on W. 2
This context is often viewed as a dangerous slippery slope,
fraught with paradoxes, or at least, quite murky, subject to
differing interpretations.
But the thrust of this work is that surprisingly minimal
principles about predication on W are needed to establish
which sentences of LPC(=) are satisfiable in domain W, and
related results.
2. TWO NOTIONS OF PREDICATION AND INTERPRETATIONS
In the context of predication on W, identity of indiscernibles states the following:
for all x,y, x = y iff for all unary predicates P on W,
P(x) ´ P(y).
This follows from singleton extensions principle:
for all x there exists a unary predicate P on W such that
"y(P(y) ´ y = x).
In mathematical and set theoretic predication, the singleton
extensions principle holds.
However, there is another concept of predication discussed by
philosophers, for which the singleton extensions principle is
doubtful.
This is the notion of predicates that are given without
reference to any particular objects, but only involving
concepts. Or one can take a linguistic tack  relations that
can be defined in a language, where language is broadly
interpreted, rather than in some particular already
formalized language.
We will use the term "general predicate" for the first
concept of predication, where the singleton extensions
principle obviously holds.
And we will use the term "pure predicate" for the second
concept of predication, where even the doctrine of identity
of indiscernibles is problematic.
What is the relationship between these two concepts, of pure
predicates and of general predicates? Obviously every 3
extension of a pure predicate is the extension of a general
predicate.
However, it does not seem that one can hope to characterize
the extensions of pure predicates in terms of the extensions
of general predicates.
One reasonable philosophical position is that the concept of
pure predication on W is fundamental, whereas any concept of
general predication on W is derived. Under this position,
general predicates are defined as cross sections of pure
predicates. We will not take this view here, but view pure
and general predication as separate, but related, fundamental
concepts.
In particular, we now have the Wp and Wg interpretations of
LPC(=). The domain is always W and the constant symbols are
always elements, but the relations and functions are pure and
general, respectively. Functions are always thought of as
univalent predicates.
A sentence in LPC(=) is called Wp (Wg) satisfiable if and
only if it is true in some Wp (Wg) interpretation. This work
deals with the determination of the Wp (Wg) satisfiable
sentences of LPC(=).
3. DOMAIN INTERPRETATION
SUBSUMED
Recall the domain interpretation. We distinguish between the
pure and the general domain interpretation. We indicate how
it is subsumed by the Wg (Wp) interpretations.
j is satisfiable in the general (pure) domain interpretation
is equivalent to the Wg (Wp) satisfiability of
($x)(P(x)) Æ j(P)
where j(P) is obtained from j by relativizing all quantifers
to the extension of P. Here P is a unary predicate symbol not
appearing in j. To accommodate constant and function symbols,
the antecedent needs to be strengthened to assert that P
holds of the constants, and holds of values of the functions
at arguments in the extension of P.
4. BASIC FORMAL THEORIES OF PREDICATION 4
In BTP (basic theory of predication), we have variables over
W (W variables), in lower case; variables over general
predicates on W, in upper case with superscript g; and
variables over pure predicates on W, in upper case with
superscript p. Also = on W, the binary function symbol < > on
W, and constant symbol 0 (an object in W).
The W terms of BTP are:
1. Every W variable and 0 is a W term.
2. If s,t are W terms then <s,t> is a W term.
The atomic formulas of BTP are:
3. s = t, where s,t are W terms.
4. Pg(t),Pp(t), where Pg is a general predicate variable, Pp
is a pure predicate variable, and t is a W term.
The formulas of BTP are:
5. atomic formulas are formulas.
6. closure under connectives.
7. closure under ",$.
Axioms and rules of BTP are:
A. Usual for L(BTP).
B. Pairing. <x,y> = <z,w> ´ (x = z Ÿ y = w).
C. Zero. <x,y> ≠ 0.
D. General Comprehension. ($Pg)("x)(Pg(x) ´ j), where j in
L(BTP) and Pg not free in j.
E. Pure Comprehension. ($Pp)("x)(Pp(x) ´ j), j in L(BTP), Pp
not free in j, no free W variables ≠ x in j, and no free
general predicate variables in j.
Let BTPp is the axioms of BTP which have no general predicate variables; BTPg no pure predicate variables.
THEOREM 4.1. Theorems of BTP in L(BTPp) = theorems of BTPp;
theorems of BTP in L(BTPg) = theorems of BTPg.
We can suitably develop arithmetic and finite sequences of W
objects as W objects in either BTPp or BTPg. There is exactly
one way to do this up to the appropriate kind of isomorphism.
The p and g developments are demonstrably equivalent in BTP. 5
Induction with respect to all formulas will be provable in
BTP.
We can also develop the satisfaction relation for any
structure whose domain is W: In BTPp for pure structures, in
BTPg for general structures.
Identity of indiscernibles:
IIS) ("x,y)(x = y ´ ("P)(Pp(x) ´ Pp(y))).
Singleton extensions principle:
SEP) ("x)($Pp)("y)(Pp(y) ´
y = x)
In BTPp, SEP implies IIS.
THEOREM 4.2. The implication IIS Æ SEP is not provable in
BTP.
5. COMPLETENESS OF PC, PC(=), PC(=,INF)
LPC = language of predicate calculus without equality, LPC(=)
= language predicate calculus with equality. Let PC and PC(=)
be the usual axioms and rules of inference.
Let PC(=,inf) be PC(=) plus:
($x1,...,xn)(x1 ≠ ... ≠ xn),
where n ≥ 1.
THEOREM 5.1. (Gödel) BTPg (BTPp) proves that a formula of
LPC(=) is general (pure) domain satisfiable iff if it is
consistent in PC(=).
We say that a sentence in PC(=) is Wg (Wp) satisfiable iff it
holds in some general (pure) W structure.
We also say that a sentence in PC(=) is Ng (Np) satisfiable
iff it holds in some general (pure) N structure.
THEOREM 5.2. (Gödel) BTPg (BTPp) proves that a sentence of
LPC(=) is Ng (Np) satisfiable iff it is consistent in
PC(=,inf). 6 The following result is proved by creating unlimited clones
of any single element.
THEOREM 5.3. BTPg (BTPp) proves that a sentence of LPC is Wg
(Wp) satisfiable iff it is consistent in PC.
Now we discuss these assertions:
Wg satisfiable = N satisfiable
Wp satisfiable = N satisfiable
Problematic. In one binary relation symbol R, look at:
R is a linear ordering.
This is Wg (Wp) satisfiable iff if there is a general (pure)
linear ordering of W.
After failing to get your candidate through the hiring
committe, you say
you can't rank order these candidates.
Maybe you could if there is a linear ordering of W.
THEOREM 5.4. The following are provably equivalent in BTPg
(BTPp):
i) there is a general (pure) linear ordering of W;
ii) a sentence in PC(=) is Wg (Wp) satisfiable iff it is
consistent in PC(=,inf).
Proof: We need only handle the p case. We argue in BTPp.
Suppose ii). Now
R is a linear ordering
is N satisfiable. Hence it is Wp satisfiable. I.e., there is
a pure linear ordering of W.
It suffices to prove that i) implies ii). The forward
direction of ii) is immediate. 7
Now suppose that there is a pure linear ordering of W. There
is a pure dense linear ordering of W without endpoints by
surrounding each point with a copy of the rationals.
Suppose j is consistent in PC(=,inf). T has an explicitly
arithmetical model M (with domain N) generated by explicitly
arithmetic Skolem functions on an infinite set of explicitly
arithmetic linearly ordered indiscernibles. We can assume
that the indiscernibles have order type the rationals.
Introduce new constants cx for each W object x. We define a
structure whose domain D consists of the closed terms in
these constants and the constant and function symbols of M.
We will use the linear ordering of W, which linearly orders
the subscripts of the new constants.
The truth value of any atomic formula will be determined by
making any order preserving assignment of indiscernibles to
the subscripts of the new constants appearing in the atomic
formula, and setting it to be the truth value of the
resulting statement in M. Because of indiscernibility, this
is independent of the choice of order preserving assignment.
We then prove by induction that any formula in the language
of M with assigned free variables (which are closed terms)
holds* in this large structure iff it holds in M after the
closed terms are changed to corresponding elements of D. Part
of the induction hypothesis is that the truth value in M so
obtained does not depend on the choice of order preserving
map.
Note the asterisk at the sticky point. The interpretation of
= is not equality, but the equivalence relation between these
closed terms, according to whether two closed terms have
equal values in M when the subscripts of the constants
appearing are mapped by an order preserving map into the
original indiscernibles.
Normally, this is remedied by calling the large structure a
weak model of the sentences true in M, and then factoring by
the equivalence relation.
The problem here is that the ensemble of equivalence classes
is not like an extension of a pure predicate  the type is
too high. 8
We need to find a pure function H from D into W such that two
terms are equivalent iff their values under H are equal. The
size of the image of H must be the same as W because the new
constants all lie in different equivalence classes.
We can then factor D by the equivalence relation using values
of H to obtain a pure structure whose domain is large; i.e.,
such that there is a oneone pure map from W into the domain.
But then we use the Shroeder Bernstein theorem in this
context, which can be proved in its pure form in BTPp. QED
THEOREM 5.5. BTP neither proves nor refutes any equality
between Wg satisfiability, Wp satisfiability, and consistency
in PC(=,inf).
6. PRINCIPLE OF SYMMETRIC ARGUMENTS
A sentence in LPC(=) is said to be existential (universal)
iff it begins with a block of zero or more existential
(universal) quantifiers followed by a quantifier free
formula.
THEROEM 6.1. It is provable in BTPg (BTPp) that every
existential sentence is Wg (Wp) satisfiable iff it is
consistent in PC(=).
It is sometimes convenient to consider the Wg (Wp) valid
formulas. A formula is Wg (Wp) valid iff its universal
closure is Wg (Wp) valid iff its negation is not Wg (Wp)
satisfiable.
The most basic sentence of PC(=) whose Wg (Wp) validity is in
question is
($x)($y)(x ≠ y Ÿ (R(x,y) ´ R(y,x))).
The general (pure) principle of binary symmetric arguments
asserts that this sentence is Wg (Wp) valid.
The general (pure) principle of symmetric arguments asserts
that for all k ≥ 1, every general (pure) predicate holds or
fails of all permutations of some ktuple of distinct
objects.
Note that this is trivial for k = 1. Also, if it holds for k
≥ 2 then it holds for k1. 9 THEOREM 6.2. The general and pure principles of symmetric
arguments are neither provable nor refutable from BTP. The
general principle does not follow from the pure principle in
BTP.
We say that a general (pure) predicate P is finite iff there
is a finite sequence x such that
("y)(P(y) Æ y is a term in x).
We say that a general (pure) predicate P is infinite iff it
is not finite.
THEOREM 6.3. BTP proves that a general predicate is finite
iff its extension is in general oneone correspondence with a
proper initial segment of N. BTP does not prove all finite
pure predicates are in pure oneone correspondence with a
proper initial segment of N.
A minimally infinite general predicate is an infinite general
predicate P such that no general predicate splits P. I.e.,for
any general predicate Q, either
i) there is a finite sequence x such that ("y)((P(y) Ÿ Q(y))
Æ y is a term in x); or
ii) there is a finite sequence x such that ("y)((P(y) Ÿ
ÿQ(y)) Æ y is a term in x).
THEOREM 6.4. BTPg proves: if $ minimally infinite general
predicate then general principle of symmetric arguments.
Converse not provable in BTP.
THEOREM 6.5. BTP does not prove or refute $ a minimally
infinite general predicate.
An absolute POI (predicate of indiscernibles) is a pure
predicate P where any pure predicate holds or fails of any
two equal length finite sequences of distinct objects from
the extension of P.
THEOREM 6.6. BTP does not prove or refute the existence of an
infinite absolute POI.
7. SATISFIABILITY OF UNIVERSAL SENTENCES 10
SYM(=) is PC(=) plus: Let k ≥ 1 and j be a formula in LPC(=)
with at most the free variables x1,...,xk. Take
($x1 ≠ ... ≠ xk)(conjunction of
(j(x1,...,xk) ´ j(xp1,...,xpk))),
conjunction ranging over all permutations p of {1,...,k}.
THEOREM 7.3. BTPp proves that every universal sentence
consistent in SYM(=) is Wp satisfiable.
Thus for universal sentences, consistency in SYM(=) implies
Wp satisfiability implies consistency in PC(=,inf).
THEOREM 7.4. The following are provably equivalent in BTPg
(BTPp).
i) the general (pure) principle of symmetric arguments;
ii) every universal sentence is Wg (Wp) satisfiable iff it is
consistent in SYM(=).
Thus if you accept the principle of symmetric arguments,
then you have completely determined the universal sentences
that are W satisfiable.
8. NECESSARY SATISFIABILITY OF UNIVERSAL SENTENCES
Another approach to Wg (Wp) satisfiability is to determine
which sentences are possibly or necessarily Wg (Wp)
satisfiable relative to BTP.
We say that a sentence j is possibly (necessarily) Wg
satisfiable over BTP iff “j is Wg satisfiable” is consistent
(provable) in BTP. The same for Wp.
THEOREM 8.1. A sentence is possibly Wg (Wp) satisfiable over
BTP if and only if it is satisfiable.
A relational sentence is one which has no constant or
function symbols.
THEOREM 8.2. A universal relational sentence is necessarily
Wg (Wp) satisfiable over BTP iff it is consistent in SYM(=).
THEOREM 8.3. A universal sentence is necessarily Wg (Wp)
satisfiable over BTP iff the assertion that it is consistent
in SYM(=) is provable in the formal system of second order 11
arithmetic. A universal sentence is necessarily Wg (Wp)
satisfiable over BTP + the true arithmetic sentences iff it
is consistent in SYM(=).
9. AN ALTERNATIVE AXIOMATIZATION
Let d1,d2,... be new constants. Let j be a formula in LPC(=)
with at most the free variables x1,...,xk and p be a permutation of (1,...,k). Take
j(d1,...,dk) ´ j(dp1,...,dpk).
We write this as SYMC(=), where C means “constants.”
THEOREM 9.1. SYM(=) and SYMC(=) prove the same formulas in
LPC(=).
So we can use the purely universal theory SYMC(=) instead of
SYM(=).
10. ALTERNATIVE DETERMINATIONS
We may deny the general (pure) principle of symmetric arguments. Then the Wg (Wp) satisfiable universal sentences will
be properly greater than those provable in SYM(=). Can we
adopt some of the general (pure) principle of symmetric
arguments without adopting all of it?
THEOREM 10.1. Let k ≥ 2. It is consistent with BTP that the
general principle of kary symmetric arguments holds but not
the general principle of k+1ary symmetric arguments.
This leads to a determination of the Wg satisfiable universal
sentences that is intermediate between consistency with
SYM(=) and satisfiability. There are a number of further
questions here that we have not investigated.
11. SATISFIABILITY OF $" SENTENCES
We characterize the Wg (Wp) satisfiable $" sentences by
allowing parameters in SYM(=).
SYM’(=) is PC(=) plus: Let k ≥ 1 and j be a formula in
LPC(=). Take
($x1 ≠ ... ≠ xk)(conjunction of 12
(j(x1,...,xk) ´ j(xp1,...,xpk))),
conjunction ranging over all permutations p of {1,...,k}.
Free variables are interpreted universally.
THEOREM 11.1. A universal sentence is consistent in SYM’(=)
if and only if it is consistent in SYM(=).
THEOREM 11.2. The following are provable in BTPp. Every $"
sentence consistent in SYM’(=) is Wp satisfiable.
THEOREM 11.3. The following are provably equivalent in BTPg
(BTPp).
i) the general (pure) principle of symmetric arguments;
ii) every $" sentence is Wg (Wp) satisfiable iff it is
consistent in SYM’(=).
12. ARBITRARY FORMULAS  MODEL THEORETIC DETERMINATION
Here we give a determination of the Wg and Wp validity of
arbitrary formulas in LPC(=) in model theoretic terms. In
particular, the sentences determined to be Wg (Wp)
satisfiable will be co r.e.
Let M be a structure, E Õ dom(M). We say E is a set of
symmetric generators of M iff every element of dom(M) is a
term of elements of E, and every permutation of E extends to
a permutation of M.
Let SYMGEN(=) be the sentences that hold in some structure
with an infinite set of symmetric generators. The type of
the structure may have to be include more symbols than those
appearing in j.
THEOREM 12.1. SYMGEN(=) is complete co r.e. An $" sentence
lies in SYMGEN(=) iff it is consistent in SYM’(=). A
universal sentence lies in SYMGEN(=) iff it is consistent in
SYM(=).
THEOREM 12.2. The following is provable in BTPg (BTPp). Every
sentence in SYMGEN(=) is Wg (Wp) satisfiable.
Let P be a general k+1ary predicate.
We say that F is a choice function for P if and only if 13
i) F is a function of the form F:Ak Æ A, where A is the
extension of a general predicate;
ii) for all x Œ Ak, if ($y)(P(<x,y>) then P(<x,F(x)>).
Principle *) asserts that every general multivariate
predicate has a choice function with an infinite set of
symmetric generators.
THEOREM 12.3. BTP does not prove or refute principle *). The
following are provably equivalent in BTPg.
i) principle *);
ii) any sentence that is Wg satisfiable lies in SYMGEN(=);
iii) a sentence is Wg satisfiable iff it lies in SYMGEN(=).
We think that *) goes against the idea behind pure
predication. So we give a weakened form of *) involving a
finite conclusion.
Let P be a general (pure) k+1ary predicate. We say that F is
a finite choice function for P if and only if
i) F is a function with domain Ak, where A is given by a
finite sequence;
ii) for all x Œ Ak, if ($y)(P(<x,y>) then P(<x,F(x)>).
We say that E is nsymmetric in F iff
i) every permutation of E extends to an automorphism of F;
ii) every term involving F and at most n occurrences of
elements of E is defined.
Principle **g) says: for all n,m every general multivariate
predicate has a choice function with a set of nsymmetric
generators of cardinality m. **p) is the same with general
replaced by pure.
THEOREM 12.4. BTP does not prove or refute principle **g) or
**p). BTPg (BTPp) proves the equivalence of:
i) principle **) for general (pure) predicates;
ii) any sentence that is Wg (Wp) satisfiable lies in
SYMGEN(=);
iii) a sentence is Wg (Wp) satisfiable if and only if it lies
in SYMGEN(=).
13. NECESSARY SATISFIABILITY OF SENTENCES 14
THEOREM 13.1. A sentence is necessarily Wg (Wp) satisfiable
over BTP iff the assertion that it has a countable model with
an infinite set of symmetric generators is provable in the
formal system of second order arithmetic.
THEOREM 13.2. A sentence is necessarily Wg (Wp) satisfiable
over BTP + the true arithmetic sentences iff it has a model
with an infinite set of symmetric generators.
14. CLASS THEORETIC INTERPRETATION OF PREDICATION ON W
The theory of classes provides appropriate models of predication on W that are familiar.
W is interpreted as the class V of all sets. 0 is interpreted as the empty set. Pairing is interpreted as a standard
pairing function in set theory. Equality is interpreted as
equality in set theory. Predicates on W are interpreted as
classes.
We have a number of options. We can allow/disallow urelements, allow/disallow the axiom of choice for sets. We then
interpret general predicates on W as arbitrary classes.
The distinction between pure and general predicates  classes
here  is best made axiomatically by disallowing set
parameters in the comprehension axiom scheme for pure
classes. ...
View
Full
Document
This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Calculus

Click to edit the document details