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
1. A finite number of basic elements (an alphabet)
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.)
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.
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:
Think now of Hilbert's two questions, to which we will add a third:
- 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.
- 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.
- 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**,
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')
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
- Rules: modus ponens, and universal instantiation.
- Decidability: can we prove this theory is decidable? A
theory is decidable if there is some mechanical method that will
tell us, for any sentence (formed using the symbols of this
theory) whether that sentence is a theorem of this theory. (A
theorem of the theory is a sentence provable from the theory
alone, without adding premises to your argument/proof.)
- Consistency: can we prove that this theory is
consistent? Note then that if we accept that this theory is
arithmetic, then if we prove this is consistent we have proven
that arithmetic is consistent. Recall that consistency means that
for no statement P can we prove both P and ¬P. In this system,
and in any standard implementation of first order logic, from P
and ¬P we can prove any sentence Q -- so, if your system is
inconsistent, it is useless because it will allow you to prove all
falsehoods. (To convince yourself, prove (¬P → (P → Q))
using the above axioms!) But, conversely, that means if we can
show that you cannot prove just any old sentence (such as, say,
the claim that ¬(1=1)) then we would know that this system was
- Completeness: can we prove this theory is complete? A
theory is complete if all the truths of the theory are provable
with the theory.
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.