Focal subgroup theorem: Difference between revisions

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In [[computability theory]], '''recursively inseparable sets''' are pairs of sets of natural numbers that cannot be "separated" with a [[recursive set]] (Monk 1976, p.&nbsp;100). These sets arise in the study of computability theory itself, particularly in relation to [[Pi01 class|&Pi;<sup>0</sup><sub>1</sub> classes]]. Recursively inseparable sets also arise in the study of [[Gödel's incompleteness theorem]].
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== Definition ==
 
The natural numbers are the set &omega; = {0, 1, 2, ...}. Given disjoint subsets ''A'' and ''B'' of &omega;, a '''separating set''' ''C'' is a subset of &omega; such that ''A'' &sube; ''C'' and ''B'' &cap; ''C'' = &empty; (or equivalently, ''A'' &sube; ''C'' and ''B'' &sube; {{overbar|''C''}}). For example, ''A'' itself is a separating set for the pair, as is ''&omega;\B''.
 
If a pair of disjoint sets ''A'' and ''B'' has no recursive separating set, then the two sets are '''recursively inseparable'''.
 
== Examples ==
 
If ''A'' is a non-recursive set then ''A'' and its complement are recursively inseparable. However, there are many examples of sets ''A'' and ''B'' that are disjoint, non-complementary, and recursively inseparable. Moreover, it is possible for ''A'' and ''B'' to be recursively inseparable, disjoint, and [[recursively enumerable]].
* Let &phi; be the standard indexing of the [[partial computable function]]s. Then the sets {{nowrap|1=''A'' = {''e'' : &phi;<sub>''e''</sub>(0) = 0 }}} and {{nowrap|1=''B'' = {''e'' : &phi;<sub>''e''</sub>(0) = 1 }}} are recursively inseparable (Gasarch 1998, p. 1047).
* Let # be a standard [[Gödel numbering]] for the formulas of [[Peano arithmetic]]. Then the set {{nowrap|1=''A'' = { #(&psi;) : PA ⊢ &psi;}}} of provable formulas and the set {{nowrap|1=''B'' = { #(&psi;) : PA ⊢ ¬&psi;}}} of refutable formulas are recursively inseparable. The inseparability of the sets of provable and refutable formulas holds for many other formal theories of arithmetic (Smullyan 1958).
 
== References ==
* {{Citation | last1=Cenzer | first1=Douglas | title=Handbook of computability theory | publisher=North-Holland | location=Amsterdam | series=Stud. Logic Found. Math. | id={{MathSciNet | id = 1720779}} | year=1999 | volume=140 | chapter=<math>\Pi^0_1</math> classes in computability theory | pages=37–85}}
* {{Citation | last1=Gasarch | first1=William | title=Handbook of recursive mathematics, Vol. 2 | publisher=North-Holland | location=Amsterdam | series=Stud. Logic Found. Math. | id={{MathSciNet | id = 1673598}} | year=1998 | volume=139 | chapter=A survey of recursive combinatorics | pages=1041–1176}}
* {{Citation | last1=Monk | first1=J. Donald | title=Mathematical Logic | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90170-1 | year=1976}}
* {{Citation | last1=Smullyan | first1=Raymond M. | author1-link=Raymond M. Smullyan | title=Undecidability and recursive inseparability | doi=10.1002/malq.19580040705 | id={{MathSciNet | id = 0099293}} | year=1958 | journal=Zeitschrift für Mathematische Logik und Grundlagen der Mathematik | issn=0044-3050 | volume=4 | pages=143–147}}
 
[[Category:Computability theory]]

Revision as of 09:21, 1 March 2014

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