# Model of a language $L$ vs. model of a theory $T$ in $L$

I am reading Just/Weese and they seem to use "model of a language $L$", for example, p. 90:

and, more disturbingly, p. 91:

Isn't this a "typo" (or perhaps sloppy writing)? If $L$ is any language and $\varphi$ any formula in $L$ then $\lnot \varphi$ is also a formula in $L$. Hence there cannot be a model of $L$.

But most likely I am missing something fundamental. So: where did I go wrong? Thanks for your help.

Edit

Perhaps this is related and the authors use theory and language as synonyms: on page 92

Well... what? Isn't ZFC the 9 or so axioms given for example here?

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Note that the axiom schema of replacement is actually an infinite list of axioms. How cool is set theory now? We can write infinitely many axioms! POW!! –  Asaf Karagila Nov 23 '12 at 12:23

It's just slightly sloppy terminology to avoid introducing nearly synonymous words.

• A model of a (first-order) language would be just a set together with appropriate interpretations of all constant, function and relation symbols.

• A model of a theory would then be a model of the language in which all the sentences/formulae of the theory are true.

You can argue that a language specifies its formulae, and therefore a model of a language should morally be a model of all formulae of that language, as per the second point, but then you have to find a way to bootstrap yourself to the second definition.

In practice this shouldn't cause too much difficulty, as whenever you come across model in Just-Weese (or elsewhere) it should be clear from context which meaning is intended.

(Sometimes structure or interpretation is used for models of the first type, reserving model for models of the second type.)

At times it's almost enough to make you wonder if the sloppiest expositions in mathematics come from the AMS's 03-XX subject classification....

Added due to edit.

I'm almost certain that Just-Weese do not use language and theory as synonyms. Consider the second paragraph of quoted text in your previous question.

And, no, ZFC does not consist of only nine axioms. In particular the Axiom Schema of Separation (sometimes called (Restricted) Comprehension, or Specification) consists of an infinite list of axioms. But I believe this comes up later in Chapter 7 of Just-Weese.

You might even see later (in Chapter 12) that there is no finite axiomatization of ZFC (assuming the consistency of ZFC).

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Note that there are also texts that speak of an "interpretation" of a language, rather than a "structure". –  Henning Makholm Nov 23 '12 at 12:35
@Henning: Thanks. I'll add it. (I admit that I was running a blank about other terms used in this context.) –  Arthur Fischer Nov 23 '12 at 12:39