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A definable set in a first-order structure $M$ is the set of tuples satisfying one first-order formula, possibly with a specified finite tuple of parameters $\mathbf{b}$ from the structure. A set definable over $B\subseteq M$ uses parameters from $B$. Parameter-free definability uses no extra named elements. The distinction matters: any singleton becomes definable if its own element is allowed as a parameter, while many singletons are not definable without parameters. Finite Boolean combinations of definable sets are definable, using logical conjunction, logical disjunction and logical negation.

A definable set has parameter-free definability when its defining first-order formula uses only the original first-order language, with no additional constants naming elements. Constants already present in the language are allowed. Every such set is invariant under all automorphisms of a first-order structure. For example, zero in the constant-free additive integers is defined by $x+x=x$, while the singleton containing one is not parameter-free definable because negation is an additive automorphism moving it. Invariance is a necessary condition in every structure, and becomes sufficient for finite sets in a countable omega-saturated model.

Here parameter-free definability is intended: parameters are not silently added to the first-order language. A subset $D\subseteq A^n$ is a definable set when some $L$-first-order formula $\phi (x_1,\dots ,x_n)$ satisfiesA definable function is one whose graph is a definable set. Thus $f:A^n\to A$ is definable when an $L$-formula $\theta (\mathbf{x},y)$ satisfiesIn particular $\mathcal{A}\models \forall \mathbf{x}\exists !y\theta (\mathbf{x},y)$. Definability over a parameter set $B$ permits parameters from $B$ in these first-order formulas, equivalently new constants naming those parameters. That is a different requirement: allowing a parameter naming an element would immediately define its singleton.

No. Interpreting the printed ${\mathbb{Z}}^{⩾}$ as the nonnegative integers, negation is an additive structure automorphism carrying the positive integer one outside that set. The set is therefore not invariant under structure automorphisms and cannot have parameter-free definability. The same argument would apply if the notation were taken to mean the strictly positive integers.