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

Revision as of 06:17, 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).

Capturing functions

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

$f$ 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]
$f$ 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 $f(m)\neq n$ then $T\vdash \neg \phi ({\overline {m}},{\overline {n}})$ .^[1]
$f$ 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]

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")^[2]
Boolos, Burgess, Jeffrey	arithmetically defines^[3]	defines (for sets), represents (for functions)^[3]
Wikipedia	arithmetically defines	this page uses "represents", but I don't think there's a standalone article for the concept

References

↑ ^1.0 ^1.1 ^1.2 Peter Smith. Godel book, p. 119, 120, 122.
↑ Goldrei. Propositional and Predicate Calculus. p. 137.
↑ ^3.0 ^3.1 George S. Boolos; John P. Burgess; Richard C. Jeffrey. Computability and Logic (5th ed). p. 199 for "arithmetically defines". p. 207 for "defines".

[smith-1] 1.0 ^1.1 ^1.2 Peter Smith. Godel book, p. 119, 120, 122.

[2] Goldrei. Propositional and Predicate Calculus. p. 137.

[boolos-3] 3.0 ^3.1 George S. Boolos; John P. Burgess; Richard C. Jeffrey. Computability and Logic (5th ed). p. 199 for "arithmetically defines". p. 207 for "defines".

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@@ Line 5: / Line 5: @@
 ==Capturing functions==
+For functions, it seems like there are at least four different strengths.
+# <math>f</math> is captured by <math>\phi(x,y)</math> iff for all <math>m,n</math> (i) if <math>f(m) = n</math> then <math>T \vdash \phi(\overline{m}, \overline{n})</math> and (ii) <math>T \vdash \exists y (\phi(\overline{m}, y) \wedge \forall v(\phi(\overline{m}, v) \to v=y))</math>.<ref name="smith">Peter Smith. Godel book, p. 119, 120, 122.</ref>
+# <math>f</math> is captured by <math>\phi(x,y)</math> iff for all <math>m,n</math> (i) if <math>f(m)=n</math> then <math>T \vdash \phi(\overline m, \overline n)</math>, and (ii) if <math>f(m)\ne n</math> then <math>T \vdash \neg \phi(\overline m, \overline n)</math>.<ref name="smith"/>
+# <math>f</math> is captured by <math>\phi(x,y)</math> iff (i) for all <math>m,n</math>, if <math>f(m) = n</math> then <math>T \vdash \phi(\overline m, \overline n)</math>, and (ii) we have <math>T \vdash \forall x \exists y (\phi(x,y) \wedge \forall v (\phi(x,v) \to v=y))</math>.<ref name="smith"/>
 ==Comparison of usage patterns==