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).

Capturing functions

For functions, it seems like there are at least four different strengths.

  1. f is captured by \phi(x,y) iff for all m,n (i) if f(m) = n then T \vdash \phi(\overline{m}, \overline{n}) and (ii) T \vdash \exists y (\phi(\overline{m}, y) \wedge \forall v(\phi(\overline{m}, v) \to v=y)).[1]
  2. f is captured by \phi(x,y) iff for all m,n, if f(m) = n, then T \vdash \forall y (\phi(\overline m,y) \leftrightarrow y = \overline n).[1]
  3. f is captured by \phi(x,y) iff for all m,n (i) if f(m)=n then T \vdash \phi(\overline m, \overline n), and (ii) if f(m)\ne n then T \vdash \neg \phi(\overline m, \overline n).[1]
  4. f is captured by \phi(x,y) iff (i) for all m,n, if f(m) = n then T \vdash \phi(\overline m, \overline n), and (ii) we have T \vdash \forall x \exists y (\phi(x,y) \wedge \forall v (\phi(x,v) \to v=y)).[1]
  5. f is captured by \phi(x,y) iff for all m,n (i) if f(m)=n then T \vdash \phi(\overline m, \overline n), and (ii) if f(m)\ne n then T \nvdash \phi(\overline m, \overline n).[2]

Comparison of usage patterns

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")[3]
Boolos, Burgess, Jeffrey (5th ed) arithmetically defines[4] defines (for sets), represents (for functions)[4]
Wikipedia arithmetically defines, i think this page uses "defines" in the expresses sense (? actually i'm not sure; this sense of "defines" seems different) this page uses "represents", but I don't think there's a standalone article for the concept

References

  1. 1.0 1.1 1.2 1.3 Peter Smith. Godel book, p. 119, 120, 122.
  2. Leary and Kristiansen. A Friendly Introduction to Mathematical Logic (2nd ed). p. 121
  3. Goldrei. Propositional and Predicate Calculus. p. 137.
  4. 4.0 4.1 George S. Boolos; John P. Burgess; Richard C. Jeffrey. Computability and Logic (5th ed). p. 199 for "arithmetically defines". p. 207 for "defines".