The definition of a structure in model theory can be summed up like this (for simplicity's sake without individual constants and functions):

Def. 1 A structure is a triple of sets $\langle A, \alpha, I\rangle$ such that:

  1. $\alpha$ is a function with rg($\alpha$) $\subseteq \mathbb{N}$ (the arity function)
  2. $I$ is a function with dom($I$) = dom($\alpha$) (the interpretation function)
  3. $I(R) \subseteq A^{\alpha(R)}$ for all $R \in$ dom($I$)

From the point of view of model theory dom($I$) = dom($\alpha$) is most important as the set of (relational) symbols, and plays a dominant role as the signature $\sigma$ of a language which is built on top of it.

If you are not so interested in model theory and the relation between language and reality, but in structures per se, you would define a (relational) structure maybe like this:

Def. 2 A structure is a pair of sets $\langle A, R\rangle$ such that

  1. $R$ is a function
  2. $(\exists \alpha \in \mathbb{N})\ R(i) \subseteq A^\alpha$ for all $i \in $ dom($R$)

It's obvious that the two definitions are equivalent, only that dom($R$) - which plays the role of the signature in the first definition - isn't so important per se and maybe only considered a index set.

What's more, if you take the position, that two coextensive relations are the same, you have no need to index the relations because they are different by themselves. The definition then simplifies to:

Def. 3 A structure is a pair of sets $\langle A, \rho\rangle$ such that

  1. $(\exists \alpha \in \mathbb{N})\ R \subseteq A^\alpha$ for all $R \in \rho$

What puzzles me is that the same set dom($I$)=dom($\alpha$) resp. dom($R$) comes into play for two seemingly unrelated reasons: on the one side to be able to define a language on top of a signature, and to be able to have coextensive but different relations on the other side. But I find it hard to make a real question out of this. Maybe like this?

Can an alternative model theory be imagined based on a definition of a structure equivalent to Def. 3 reflecting the assumption that coextensive relations are equal?


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