A note about Theories

A Note about Theories
Most of the interesting results that we are going to consider later in the course require that a theory be a clearly defined kind of object. For that reason, we need to assume the following. Whenever we refer to a "theory" we will mean a thing that has the following features:
1. A finite number of basic elements (an alphabet)

2. Finite set of formation rules that tell us which combinations of these elements are well-formed formulas (this is like the most basic level of the grammar, and includes statements which are obviously not true -- just as it is grammatical in English to say false things like "Lake Ontario contains no water," and also weird things like "The color Nixon is taller than Smith winks," it is grammatical though false to say in a standard axiomatization of arithmetic something that means "1 is not equal to 1").

3. A finite set of axioms or axiom schemas (these are assumed statements, or schemas out of which you can derive assumed statements)

4. A finite set of inference rules.
We call anything provable in a theory (without using extra assumptions or premises) a theorem of that theory. (Things are proved using the application of rules to the axioms to produce new sentences that are not axioms.)

Sometimes, we will call a language the total set of theorems of the theory (this is common in computer science, not so much in logic). Thus, the language is all the statements that can be derived from a theory. All of the theorems of arithmetic is all of the statements of arithmetic you can derive from basic arithmetic theory, and we can call this the "language of arithmetic."

Here is an example of most of a theory for arithmetic (I follow Smullyan and Mendelson); this kind of theory is sometimes called Peano Arithmetic, after a mathematician who first proposed a similar list of axioms:
  1. Elements. We'll hide these for now, since we are later going to rephrase them in a different way. But these include: ', 0, +, v, (, and ). More on this later.
  2. Formation rules: here I'll just give some examples. Any two terms flanking a "+" yields a new term; any two terms flanking a "=" makes a sentence; any sentence with a "¬" in front of it is a sentence; any two sentences flanking a "→" form a sentence; etc.
  3. Axioms (the first 5 constitute standard first order logic, and the rest are a version of Dedekind's axiomatization of arithmetic; assume P, Q, R are any sentences or predicates, so that P(x) is a well formed formula in which at least x appears as a free term). v1 is short for v*, and v2 is short for v**, and etc.
    A1: (P → (Q → P))
    A2: ((P → (Q → R)) → ((P → Q) → (P → R))
    A3: ((¬Q→¬P) → ((¬Q → P) → Q))
    A4: (Av1)P(v1) → P(t), where t is any term that replaces the free occurences of v1 in P.
    A5: (∀v1)(P → Q) → (P → (∀v1)Q) if P contains no free occurences of v1.
    A6: (v1=v2 → (v1=v3 → v2=v3))
    A7: (v1=v2 → v1'=v2')
    A8: ¬(0=v1')
    A9: (v1'=v2' → v1=v2)
    A10: (v1+0) = v1
    A11: (v1+v2') = (v1 + v2)'
    A12: (v1 * 0) = 0
    A13: (v1 * v2') = ((v1 * v2) + v1)
    A14: For any well formed formula P(x) of the theory, P(0) → (∀v(P(v) → P(v')) → ∀v(P(v)))

    For A6-A13, one might expect a quantifier, but that is usually implicit in the metalanguage. We are used to that: it means think of v1 as any arbitrary number.

  4. Rules: modus ponens, and universal instantiation.
Think now of Hilbert's two questions, to which we will add a third:

The axiomatization used above follows that in Mendelson, Introduction to Mathemetical Logic, 3rd Edition, 1987. The elements of the language are from Smullyan Godel's Incompleteness Theorems.