(You really want to know more about mathematical logic, or even model theory, to understand these concepts; this is just a short explanation which should help you keep your head above the water in the NSA series)
Models and languages are intertwined; you should probably read NSA: What's a language?
A model gives the connection between the language of a formal language and the objects of the world it represents. The model consists of 4 parts:
- A set of objects. Quantifiers ("for all" and "there exists") in the formal language will range over this set.
- For each constant of the language, an assignment of an object to the constant. It may well happen that not all objects of the model are assigned to constants! Some of these objects may still be reached, by combining constants and functions; others might not be, and the only way to reach them is through a quantifier. For instance, in the language of real numbers alluded to in here, the model may well assign "0" to 0, "1" to 1, and "e" to e. Then again, it could just as well assign all 3 to -17.
- For each predicate name of the language, an assignment of an actual predicate to the name. The "natural" assignment for the examples would be the "is less than" relation to "<", and the relation "is an integer" to "Integer". Again, there is nothing (yet) obligating us to make these "correct" assignments.
- For each function name of the language, an assignment of an actual function to the name. Thus, we could assign addition and multiplication to "+" and "*", and the sine function to "sin". Or we could make multiplication "+", subtraction "*", and the square function "sin".
Given a model, we can assign a truth value
to any sentence
. For example, the sentence "1*1=1" is true
in any model of the real numbers assigning 1 to "1" and multiplication to "*" (it's also true if the model assigns 0 to "1" and addition to "*"!); it is false in a model assigning -17 to "1" and multiplication to "*" (since it would then be stating that multiplying (-17) by itself yields -17, which is false
To specify a model more precisely, we can demand that it satisfy some set of sentences known as axioms. Thus, we consider M a model of the natural numbers if it satisfies the infinite set of axioms known as the Peano axioms (infinitely many axioms, and a language giving a name for every predicate on the natural numbers, are required to express the axiom of induction; one for each predicate). We consider G a model of plane geometry if it satisfies Euclid's axioms (and a few more axioms discovered in the 19th century). If we drop the parallel postulate, many more models are possible, giving G's which are models of non-Euclidian geometry.
Model theory deals with the characterisations of models in terms of the sentences which are true in them. The model should be thought of as an interpretation for any sentence in the formal language.