Immerman and Vardi's theorem shows FO(LFP) captures P over ordered structures, demonstrating that FO with least fixed-point extension does reach P without abandoning first-order syntax.

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Captures(what the theorem demonstrates): In logic, to 'capture' a set of problems means to be able to express or solve exactly those problems and no others.
FO (First-Order Logic)(the main subject being discussed in the statement): A formal system for reasoning that can make statements about individual things and their properties, but cannot directly talk about properties themselves the way higher-level systems can.
FO(LFP)(Descriptive complexity; extends expressive power of first-order logic): The extension of first-order logic with new relation symbols LFP_{ψ(R,x-vec)} for each formula ψ(R,x-vec) in which the relation variable appears only positively, with atomic formulas LFP_{ψ(R,x-vec)}(t-vec) interpreted as holding iff t-vec is in the least fixed point of the monotone operator induced by ψ.
First-Order Syntax(what the enhancement preserves): The grammatical rules and symbols used to write statements in first-order logic, which restrict what kinds of statements you can make.

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