When we say "cardinality of first order language L" and "cardinality of a structure or model" what we are meaning?

Question

I ask about for what set we are referring for these cardinalities?

Answer 1

The cardinality of a structure means the cardinality of the underlying set of the structure. This is exactly what you should expect from ordinary mathematical usage: the cardinality of a group or a field or a topological space is the cardinality of its underlying set.

The cardinality of a language means the cardinality of the set of symbols in the language.

There is a slight ambiguity here: some people take "language" to mean a set of constant, function, and relation symbols, while others take "language" to mean the set of all formulas built from those symbols. With the latter convenition the cardinality of the language would be the cardinality of this set of all formulas.

But there's not much difference between these two conventions: given a set of symbols of cardinality $\kappa$, the set of first-order formulas built from those symbols has cardinality $\max(\kappa,\aleph_0)$, which is equal to $\kappa$ when $\kappa$ is infinite, or $\aleph_0$ when $\kappa$ is finite.

Answer 2

The cardinality of the structure is the cardinality of the domain set of the structure.

The cardinality of the language can mean one of two things:

1. The cardinality of the set of non-logicial symbols
2. The cardinality of the set of formulas

In my experience, usually what is meant is the first. But in standard first-order logic, these will be the same except in the case where the first is finite, in which case the second will be $\aleph_0.$ If the first is an infinite cardinal, then they are the same.