SUO: *Date 20 Apr 2002 -- Theory Bleary
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Theory Bleary
I have been making a dedicated effort to take this
first-order logic, formal semantics, model theory
approach to our so-called real existence seriously,
and maybe it's just the end of the week let-down,
but there just seems to be some definite problems.
The kicker came when I ran into the first-order
description of 'group theory' as it comes to be
embedded within these layers of formal batting.
Here is an abridged version of what it takes to
say what a 'group' is, as a model of a certain
first order theory -- patience, it takes a bit:
Here is what it takes to say what a 'sentence' is:
| 1.3. Languages, Models, and Satisfaction
|
| To formalize a language $L$, we need the
| following 'logical symbols' (see the
| corresponding development for $S$
| in Section 1.2):
|
| 1. Parentheses. '(' and ')'
|
| 2. Variables. v_0, v_1, ..., v_n, ...
|
| 3. Connectives. '&' (and), '~' (not)
|
| 4. Quantifier. '`A`' (for all)
|
| and one binary relation symbol '=' (identity).
|
| We assume, of course, that no symbol in $L$
| occurs in the above list. Certain strings
| of symbols from the above list and from $L$
| are called 'terms'. They are defined as
| follows:
|
| 1.3.1. [Definition of a 'term' of $L$].
|
| 1. A variable is a term.
|
| 2. A constant symbol is a term.
|
| 3. If F is an m-placed function symbol
| and t_1, ..., t_m are terms, then
|
| F(t_1 ... t_m) is a term.
|
| 4. A string of symbols is a term
| only if it can be shown to be
| a term by a finite number of
| applications of (1, 2, 3).
|
| The 'atomic formulas' of $L$ are strings of the form given below:
|
| 1.3.2. [Definition of an 'atomic formula' of $L$].
|
| 1. t_1 = t_2 is an atomic formula,
| where t_1 and t_2 are terms of $L$.
|
| 2. If P is an n-placed relation symbol
| and t_1, ..., t_n are terms, then
|
| P(t_1 ... t_n) is an atomic formula.
|
| Finally, the 'formulas' of $L$ are defined as follows:
|
| 1.3.3. [Definition of a 'formula' of $L$].
|
| 1. An atomic formula is a formula.
|
| 2. If p and q are formulas, then
|
| (p & q) and (~p) are formulas.
|
| 3. If v is a variable and p is a formula, then
|
| (`A`v)p is a formula.
|
| 4. A sequence of symbols is a formula
| only of it can be shown to be a
| formula by a finite number of
| applications of (1, 2, 3).
|
| At this point we assume that the reader has enough experience in first-order
| predicate logic to continue the development on his [or her] own. In particular,
| we leave it to him [or her] to decide on the notions of 'subformulas', 'free' and
| 'bound' occurrences of a variable in a formula, and to give a proper definition
| (based on definitions 1.3.1, 1.3.3) of 'substitution' of a term for a variable
| in a formula.
|
| A 'sentence' is a formula with no free variables.
|
| Chang & Keisler, 'Model Theory', pages 22-23.
Here is what it takes to say what a 'theory' is:
| 1.4. Theories and Examples of Theories
|
| A (first-order) theory T of $L$ is a collection of sentences of $L$.
| T is said to be 'closed' iff it is closed under the |= relation.
| In view of Table 1.3.1, this is the same as requiring that T
| be closed under |- . Since theories are sets of sentences
| of $L$, we may apply the expressions:
|
| a model of a theory,
|
| consistent theory,
|
| satisfiable theory,
|
| as introduced in Section 1.3.
|
| A theory T is called 'complete' (in $L$) if and only if its set of
| consequences is maximal consistent. If T is a theory of $L$, with
| $L$ c $L$’ and $L$ =/= $L$’, then T is not a closed theory of $L$’.
| On the other hand, it is easy to see that if $L$’ c $L$, then the
| 'restriction' of a closed theory T to $L$’, in symbols T | $L$’,
| is always a closed theory of $L$’. T is a 'subtheory' of T’ iff
| T c T’. If T is a subtheory of T’, then T’ is an 'extension' of T.
|
| A 'set of axioms' of a theory T is a set of sentences with the
| same consequences as T. Clearly, T is a set of axioms of T, and
| the empty set is a set of axioms of T if and only if T is a set
| of valid sentences of $L$. Every set of sentences !S! is a set
| of axioms for the closed theory T = {p : !S! |= p}. A theory T
| is 'finitely axiomatizable' iff it has a finite set of axioms.
|
| The most convenient and standard way of giving a theory T is by
| listing a finite or infinite set of axioms for it. Another way
| to give a theory is as follows: Let $A$ be a model for $L$;
| then the 'theory of' $A$ is the set of all sentences which
| hold in $A$. The theory of any model $A$ is obviously
| a complete theory.
|
| Historically, the importance of theories stems from the following
| two facts. Once the axioms of a theory are given, then by using
| the relation |- we can find out, in a syntactical manner, all the
| consequences of T. On the other hand, by using the satisfaction
| relation, we can also study all the models of T.
|
| By the extended completeness theorem, these two approaches
| give basically the same results about consequences of T.
| However, owing to the fact that models of T also have
| non-first-order properties, such as isomorphism,
| submodels, extensions, plus many others, the
| second approach leads to the field now
| known as model theory.
|
| We shall give in the rest of this section some examples of theories
| and their models to show the intimate connections that model theory
| has with other branches of mathematics. In each example we describe
| a closed theory by a set of axioms. Some classical results will be
| stated without proof.
|
| Chang & Keisler, 'Model Theory', pages 36-37.
Here is what it takes to say what a 'group' is:
| 1.4.6. Example. [The Theory of Groups].
|
| Let $L$ = {·, 1},
|
| where · is a 2-placed function symbol
| and 1 is a constant symbol.
|
| The theory of 'groups' has the following axioms:
|
| 1. Associativity
|
| (`A`xyz) x · (y · z) = (x · y) · z
|
| 2. Identity
|
| (`A`x) x · 1 = x
|
| (`A`x) 1 · x = x
|
| 3. Inverses
|
| (`A`x)(`E`y) x · y = 1 & y · x = 1
|
| A model <G, ·, 1> of this theory is a 'group'.
|
| [Modified slightly from what appears in C&K].
| Chang & Keisler, 'Model Theory', pages 39-40.
Group theory is a beautiful and fascinating subject.
I pursued it intensely, more or less, for a decade
or so, and know people who spend their lives at it,
and don't get me wrong, everything C & K say is true,
in a modus punens detetched observer impoverished way,
but there is no trace left in the above formaldehyde
that would tell you why it is interesting, much less
how to go about actually doing it. And the impression
given that one chases turnstile_1 |- and turnstile_2 |=
from the initial axioms to any old sentences that follow,
or even picks one fixed model G after another for study,
is just plain ludicrous and misleading.
This is the price we pay for buying those beans
about inquiry reducing to deductive logic and
proof within the limits of syntax alone.
Next week maybe I'll make an attempt to figure out
what exactly has gone wrong here and how all of the
motivation got drained out of this theory of theory.
Jon Awbrey
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