First-order languages, signatures, and how we build mathematical worlds
Every mathematical world has two layers: a list of symbols, and a structure that gives each symbol a meaning. Build one yourself, then watch the same set change character as the symbols change.
Add elements, declare a relation symbol with an arity, and add tuples to it. Watch the graph draw an arrow for each pair you put in a relation.
Build a first-order structure by defining a domain of elements, declaring relation symbols with arities, and specifying which tuples belong to each relation.
M = ({a, b, c}; Rᴹ = {(a,b), (b,c)})
Model theory begins with a deceptively simple question: how do we describe a mathematical world? The answer involves two layers. First, we fix a signature (or language) -- a collection of constant symbols, function symbols, and relation symbols, each with a specified arity. Then we build a structure: a nonempty domain of elements together with an interpretation that assigns a concrete meaning to every symbol in the signature.
This separation of syntax (the signature) from semantics (the structure) is the engine that drives all of model theory. The same signature can be interpreted in wildly different domains, and the same domain can carry many different signatures. A substructure arises when we restrict to a subset of the domain that is "self-contained" -- closed under all the operations and inheriting all the relations of the parent structure.
Key insight: A structure is more than just a set of elements -- it is a set together with specific interpretations of every symbol in the signature. Two structures with the same domain but different relation interpretations are genuinely different mathematical objects, even though they share the same underlying set.
Pick structures to compare side by side: the same small set as a group, a ring, a linear order, a graph, and more. Notice how each signature gives the set a different shape.
The same underlying set can carry completely different mathematical structures depending on which signature we impose. Select up to three structures to compare side by side.
The integers modulo 3 under addition
The natural ordering on three elements
A simple undirected graph (cycle)
Shared domain elements: {0, 1, 2} -- the same set, but the mathematical structure is entirely determined by which symbols we interpret on it.
Group: sigma = ({e}, {+}, empty)
Linear Order: sigma = (empty, empty, {<})
Graph: sigma = (empty, empty, {E})
Key insight: The signature determines what questions you can even ask about a structure. In the language of groups you can ask "is every element its own inverse?" but in the language of graphs that question is meaningless. Model theory studies what is expressible in a given language and what properties are preserved across different interpretations.
Click elements of the group to select a subset and see whether it is closed under addition. When it isn't, the explorer shows the pair whose sum escapes. Try subsets whose size does not divide the group order.
Select a subset of elements by clicking on them. The explorer checks whether your subset is closed under the group operation and thus forms a substructure (subgroup).
This IS a substructure (subgroup).
The subset {0, 2, 4} contains the identity, is closed under +, and every element has its inverse in the subset.
Key insight: Substructures preserve the "shape" of the parent structure on a smaller scale. In a group, the substructures are exactly the subgroups. In Z/nZ, Lagrange's theorem tells us the size of every subgroup must divide n -- try to find subsets whose size does not divide the group order, and you will see the closure check fail every time.
Spot a mistake or have an idea?