Is there a name for the construction of objects as n-tuples with axioms from a set of elements?

Question

Disclaimer: I'm not a math student, I'm studying physics.

Sometimes when I see mathematical structures defined they are just introduced in words, but sometimes they are written as an "n-tuple" with axioms (slightly contrived name since I'm not sure what the correct word is), as in the case of a group, $$(G,\cdot)$$ plus the list of axioms. Or in the case of a vector space, $$(V,+,\cdot)$$ plus axioms. Is there a name for constructing objects explicitly like this? I find it really helpful and makes it clear how different objects are related.

Answer 1

Formally speaking, this is called a structure. A structure is a triple $(A, \sigma, I)$ consisting of a domain $A$, a signature $\sigma$, and an interpretation function $I$. A signature is a tuple $\sigma = (S_\text{fun}, S_\text{rel}, \text{ar})$ consisting of a set of function symbols $S_\text{fun}$, a set of relation symbols $S_\text{rel}$, and a function $\text{ar} : S_\text{fun} \cup S_\text{rel} \rightarrow \mathbb{N}$ that assigns an arity to each symbol. The interpretation function $I$ assigns a function to every function symbol and a relation to every relation symbol. That is,

$$I : \left(\prod_{s \in S_\text{fun}} A^{\text{ar}(s)} \rightarrow A \right) \times \left(\prod_{s \in S_\text{rel}} \mathcal{P}(A^{\text{ar}(s)})\right)$$

where $\mathcal{P}$ is the powerset.

For example, for a group we have $S_\text{fun} = \{\mathtt{1}, \mathtt{\times}, \mathtt{^{-1}}\}$ and $\text{ar} = \{(\mathtt{1}, 0), (\mathtt{^{-1}}, 1), (\mathtt{\times}, 2)\}$.

Answer 2

The other answer focuses only on structures in first-order logic, which is not a good idea. What you are looking at in general are called axiomatizations, and there are many widely-used structures whose axiomatizations are not first-order. For example, the standard axiomatization of the reals is a second-order axiomatization, because the supremum axiom is a second-order axiom. Note that this depends on the ambient foundational system, because the supremum axiom can only be used on sets of reals, and what sets of reals exist depends on the axioms of the foundational system. In general, a structure with a second-order axiomatization does not pin down what sets of elements exist, and so are not as 'absolute' as first-order axiomatizations. It turns out that there cannot be a first-order axiomatization of the reals, as briefly touched on in this post.

Some widely-used mathematical structures that require second-order axiomatization include:

1. Topological space: A collection of sets that is closed under (binary) intersection and arbitrary union. [Arbitrary union is second-order.]

2. Measure space: A measure space is a σ-algebra $S$ with a non-negative countably-additive real function on $S$. A σ-algebra is a collection of subsets of some set $X$ that is closed under complement relative to $X$ and countable union. [Countable additivity/union are 2nd-order.]

3. Complete metric space: A complete metric space is a metric space that is 'closed' under limits of Cauchy sequences. [Cauchy sequences are second-order.]

4. Ordinals: The class of ordinals is the intersection of all classes that are closed under successor (i.e. $x↦x\cup\{x\}$) and arbitrary (set-sized) union. [Arbitrary union is second-order. This example is best expressed in a class theory such as NBG.]

The basic ingredient in any axiomatization is a formal system (which in mathematics is almost always based on classical first-order logic) paired with rules or axioms about the subject matter. See this post for more intuition about what axioms mean for first-order structures. A first-order theory is completely defined by FOL plus the axioms for that theory, but sometimes it is more intuitive to have rules than axioms, such as the induction rule:

Induction: Given any property $Q$ on $\mathbb{N}$, if you deduced $Q(0)$ and $∀k∈\mathbb{N}\ ( Q(k)⇒Q(k+1) )$ (in some context), then you can deduce $∀k∈\mathbb{N}\ ( Q(k) )$ (in the same context).

Compare with the induction axiom schema (i.e. list of axioms) for first-order set theories with $⟨\mathbb{N},0,1,+⟩$ defined:

Induction schema: Τhere is the axiom $φ(0) ∧ ∀k∈\mathbb{N}\ ( φ(k)⇒φ(k+1) ) ⇒ ∀k∈\mathbb{N}\ ( φ(k) )$, for each formula $φ$.

While both are equivalent, it is beneficial to understand induction as the rule as explained in this post.

Also, as mentioned earlier, second-order axiomatizations are not actually complete in the sense that they need to be part of a larger foundational system. However, categoricity theorems (such as the uniqueness of the reals up to isomorphism) show that to some extent second-order axiomatizations can in fact be sufficient. For example, the axiomatization of the reals plus some relatively weak set-theoretic axioms suffice to prove that there is a model of the real axioms that is isomorphic to every model of the real axioms. This implies that, within any set-theoretic world satisfying these, there is essentially only one model of the real axioms. Of course, different set-theoretic worlds may have non-isomorphic reals, just as different models of a first-order theory can be non-isomorphic.

As for the tuple format, that is merely an artifact of how we can encode the notion of a first-order theory in a foundational system that supports tuples and collections and functions and relations. Since each first-order theory describes what is true for some structures, any first-order structure satisfying that theory can be described by its domain (the objects in that structure) and what each symbol in the language means in that structure. Practical axiomatizations usually have finitely many symbols, which is why you usually see the domain and symbol interpretations listed as a tuple, conventionally putting the domain first. General first-order theories/structures can have arbitrary symbols, even uncountably many, in which case you cannot use a simple tuple but must use an interpretation function to map each symbol to its meaning in that structure.