Computability and Logic

12
Models
A model of a set of sentences is any interpretation in which all sentences in the set are
true. Section 12.1 discusses the sizes of the models a set of sentences may have (where
by the size of a model is meant the size of its domain) and the number of models of a
given size a set of sentences may have, introducing in the latter connection the important
notion of isomorphism. Section 12.2 is devoted to examples illustrating the theory,
with most pertaining to the important notion of an equivalence relation. Section 12.3
includes the statement of two major theorems about models, the Löwenheim–Skolem
(transfer) theorem and the (Tarski–Maltsev) compactness theorem, and begins to illustrate
some of their implications. The proof of the compactness theorem will be postponed
until the next chapter. The Löwenheim–Skolem theorem is a corollary of compactness
(though it also admits of an independent proof, to be presented in a later chapter, along
with some remarks on implications of the theorem that have sometimes been thought
‘paradoxical’).
12.1 The Size and Number of Models
By a model of a sentence or set of sentences we mean an interpretation in which the
sentence, or every sentence in the set, comes out true. Thus Ɣ implies D if every
model of Ɣ is a model of D, D is valid if every interpretation is a model of D, and Ɣ
is unsatisfiable if no interpretation is a model of Ɣ.
By the size of a model we mean the size of its domain. Thus a model is called
finite, denumerable, or whatever, if its domain is finite, denumerable, or whatever.
A set of sentences is said to have arbitrarily large finite models if for every positive
integer m there is a positive integer n ≥ m such that the set has a model of size n.
Already in the empty language, with identity but no nonlogical symbols, where an
interpretation is just a domain, one can write down sentences that have models only
of some fixed finite size.
12.1 Example (A sentence with models only of a specified finite size). For each positive
integer n there is a sentence I n involving identity but no nonlogical symbols such that I n
will be true in an interpretation if and only if there are at least n distinct individuals in
the domain of the interpretation. Then J n =∼I n+1 will be true if and only if there are
at most n individuals, and K n = I n & J n will be true if and only if there are exactly n
individuals.
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