User:IssaRice/Computability and logic/Expresses versus captures

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The expresses versus captures distinction is an important one in mathematical logic, but unfortunately the terminology differs wildly between different texts. The following table gives a comparison.

  • Expressing is done by a language. There is only one form of expressing; I think this follows from the wikipedia:Law of excluded middle.
  • Capturing is done by a theory or by axioms. There are two forms of capturing: strong capture (corresponding to deciding), and weak capture (corresponding to recognizing, or semi-deciding).
Text "Expresses" "Captures"
Peter Smith. Godel book (see especially footnote 9 on p. 45) expresses captures
Leary & Kristiansen defines represents
Goldrei defines (but the book also uses "represents")[1]
Boolos, Burgess, Jeffrey arithmetically defines[2] defines (for sets), represents (for functions)[2]
Wikipedia arithmetically defines


  1. Goldrei. Propositional and Predicate Calculus. p. 137.
  2. 2.0 2.1 George S. Boolos; John P. Burgess; Richard C. Jeffrey. Computability and Logic (5th ed). p. 199 for "arithmetically defines". p. 207 for "defines".