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A COMPLETE THEORY OF EVERYTHING:
satisfiability in the universal domain
Harvey M. Friedman
October 10, 1999
[email protected]
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. Nonempty sets.
2. N.
3. Infinite sets.
4. Fixed infinite set.
5. 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.
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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 indiscerni
bles 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.
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 Fall '08
 JOSHUA
 Math, Calculus, Equivalence relation, Firstorder logic, Axiom of choice, BTP

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