Dear Folks:

I would like to set down some general thoughts on the structure of the proofs in our scenarios collection as elaborated so far. These are prompted by review of and reflection on a sampling of the proofs. It seems to me that these can be seen as centering around three basic kinds of internal mini-strategy:

(i) Instantiation: identification of the set-theoretic objects whose special properties 'make' the proof.

(ii) Simplification: rewriting these objects, or other objects derived from them or properties of such objects, in relatively routine ways. For example, Skolemization of one or more quantifiers leads to new constant objects c having the asserted properties; if the intersection of two sets is non-null there exists an object belonging to both.

(iii) Combinatorial search (this is internal to our proof methods, and determines their 'reach'): since simplification can generally follow many paths, Combinatorial search tries to follow as many of them as possible in the hope of making a significant connection. 

In the fractal landscape of formulae in which proof operates, Instantiation identifies the high passes that must be crossed to get from hypotheses to QED. Simplification follows paths down from these peaks in various directions, attempting to set up some related section of said path. Combinatorial search explores low passes in the neighbohod of valleys found by simplification, in an atttempt to find local connections.

How can we expect most instantiations to be expressed, and what makes them significant?

This will happen in a variety of ways.

Objects can be Instantiated either

(i) Explicitly, by writing a setformer (using the expressive power of set theory directy)

(ii) By use of a theory, e.g. theory of sigma or of equivalence classes.

(iii) By minimization using 'arb'. This is the principle of induction: if there is an n having a property P, there there is an m having a property P and also smallest among all such n.

(iv) By minimization among real numbers. This is the continuous analog of (iii): if a real x having a property Pexists, and if the set of such x is bounded above, it has a least upper bound. Often this is used in the form of a a 'path must cross' principle.

(v) By the principle that a continuous function on a compact set must attain its maximum. This is a kind of multidimensional generalization of (iv).

(vi) By the Calculus principle that a differentiable function f higher in the middle of an interval than at its ends admits a maximum x which also has the propetrty that f'(x) = 0.

(vii) By the multidimensional or infinite-dimesional generalization of (vi), wcich is often expressed using Lagrange's principle. 

(viii) By some counting argument, or its continuous analog, which establishes that a set must be nonempty and so have a member.

Other arguments are essentially algebraic rather than instantiating, and typically depend on favorable cancellations.

Jack

