Difference between revisions of "User:IssaRice/Computability and logic/Expresses versus captures"

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! Text !! "Expresses" !! "Captures"
 
! Text !! "Expresses" !! "Captures"
 
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| Peter Smith. Godel book || expresses || captures
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| Peter Smith. Godel book (see especially footnote 9 on p. 45) || expresses || captures
 
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| Leary & Kristiansen || defines || represents
 
| Leary & Kristiansen || defines || represents

Revision as of 01:34, 7 February 2019

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

References

  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".