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]]

Latest revision as of 16:04, 26 November 2014

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Some also use the liner as a size aid, for example BMX helmets can come as one-size, with optional liner kits to change the helmets size. Mulberry Gap Mountain Bike Get-A-Way is located 12 miles from downtown Ellijay in the Chattahoochee National Forest with some of the best single track mountain bike trails including the Georgia Pinhoti trail system. But when it comes to mountain biking it is not enough that you own a bike. Another way to classify brakes is by mounting style. These are stable forks whose weights are directly in proportion to their durability.

I would wake up at 3 o'clock in the morning to make calls to America. Or get the type motor that senses when you are pedaling harder and switches on automatically. If the metal tubing below the paint has become exposed, then touch this up with a dab of enamel paint, using a very fine brush. There is no shortage of videos of mountain bike crashes from around the world, with many riders with appropriate safety equipment escaping serious injury, while others were not that fortunate. The biker will encounter numerous obstacles along the way which will make his trajectory an extreme adventure.

I ran with Hale and we also tried this method, it was effective. Instability: You are much more likely to lose your balance while on a big bike. Are you going to be going up and down hills, across flat terrain, on cement, or through rough and rocky areas. The Red loop has a lot of climbing and some very tight turns. They will provide you with a honest price, yet be ready to pay in between $650 and $4,000 on dual suspensions and in between $470 and $670 for hardtails.

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