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In first-order logic, a set $\tau$ of terms is the smallest set of strings (finite sequences of symbols) over the sets of variables, constants, and function symbols such that:

  • Any variable is a term.
  • Any constant is a term.
  • if $f^n$ is a function symbol of arity $n$, and $t_1,t_2,...,t_n$ are terms then $f^nt_1t_2...t_n$ is also a term.

What does the smallest set mean here? Is it really possible for more than one set satisfying the above properties to exist? If yes, an example would be great.

Thank you!

stoic-santiago
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1 Answers1

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As Kevin P. Barry comments, this is exactly to rule out unwanted objects. For example, consider the following definition of the set $E$ of even integers:

$E$ is the smallest set containing $0$ and closed under the maps $x\mapsto x+2$ and $x\mapsto x-2$.

The "smallest set" condition there is crucial: if we omit it, there are lots of sets satisfying the definition (e.g. $\mathbb{R}$ itself, $E\cup\{\pi+2k: k\in \mathbb{Z}\}$, and so forth). Another famous example of a definition of this form is one of the standard definitions of the natural numbers in set theory:

The set of finite ordinals is the smallest set containing $\emptyset$ and closed under the map $x\mapsto x\cup\{x\}$.

These show up all over the place in mathematics. (A useful term here is "inductive definition" or "recursive definition.") Generally we have the following useful heuristic:

Most of the time, "the smallest set $X$ such that [stuff]" is exactly the intersection of all the sets such that [stuff].

In fact, this is usually how we prove that a smallest such set exists - that's something we can't ignore! (E.g. "The smallest infinite set of natural numbers" isn't a thing.)

Incidentally, I've said a bit more about this in the specific context of formal languages (as in your OP) here.

Noah Schweber
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