Using an Infinite Universe in a Formal Langauge Structure

Question

I'm working through Leary and Kristiansen's "a friendly introduction to mathematical logic". When I get to page 23, they have introduced the idea of a formal language, along with the concepts of atomic and composite terms, and similarly formulae and sentences, and induction on complexity. On page 23 they introduce $\mathcal L$-structures, and at this point I get confused.

They have the langauge of number theory, $\mathcal L_{NT}:=\{0,S,+,\cdot,<,E\}$, and they introduce the associated $\mathcal L$-structure $\mathfrak{N} :=(\mathbb{N},0,S,+,\cdot,<,E).$ I have some philosophical difficulties with introducing $\mathbb{N}$ at this point.

My understanding was that we were going to go through building up to different mathematical formalism necessary to be able to construct $\mathbb{N}$. So why does it make sense to just introduce this set at this point as though it's something that we already have? We haven't yet assigned any meaning to the symbols in $\mathcal L_{NT}$, (ie. as yet no Peano axioms). My understanding is that it doesn't make sense to talk about an infinite set unless we have a definition for what $<$ means. I think I would be happy if we had put a finite set in here as the universe, but to me putting in $\mathbb{N}$ results in some kind of circular definitions.

Can anyone clear up my confusion on this? I'm not even really sure if my question is mathematical in nature or philosophical.

Answer 1

My understanding was that we were going to go through building up to different mathematical formalism necessary to be able to construct $\mathbb N$.

It is not so; we assume the existence of a structure: a domain of mathematical objects with the relevant properties. The structure $\mathfrak{N}$ with domain $\mathbb N$ is the usual domain of natural numbers.

We set-up a formal language that we will use to "talk about" the mathematical object.

More specifically, we have a previous understanding of the relation "less than" between numbers, and we use it to interpret the binary relation symbol of the language: $<$.

See page 22:

Let us return to the language $\mathcal L_{\text{NT}}$ of number theory. [...] We now want to discuss the possible mathematical structures in which we can interpret these symbols, and thus the formulas and sentences of $\mathcal L_{\text{NT}}$.

It is certainly the case that you know an interpretation for these symbols. The point of this section is that there are many different possible interpretations for these symbols, and we want to be able to specify which of those interpretations we have in mind at any particular moment.

See page 23: "Def.1.6.1. Fix a language $\mathcal L$. An $\mathcal L$-structure $\mathfrak A$ is a nonempty set $A$, called the universe of $\mathfrak A$, together with..."

The semantical clauses of the definition assign meaning to symbols of the language using the domain of mathematical objects with their properties: this is an interpretation.

The interpretation assigns to the symbols some objects7 of the domain: constant symbols are mapped to designated elements of the domain, unary predicate symbols to subsets of the domain, binary predicate to relation on the domain, and so on.

Thus, we can re-phrase the paragraph preceding that definition as follows: "determining the structure under consideration, we decide how we wish to interpret the symbols of the language; in this way, we have a way of talking about the truth or falsity sentences, like e.g. $(\forall v_1)(v_1 < S(v_1))$."