Torsion group
Mathematical logic: Precisions, and lexical modification from type to sort (more adequate in that context).
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An interesting property of periodic groups is that the definition cannot be formalized in terms of [[first-order logic]]. This is because doing so would require an axiom of the form |
An interesting property of periodic groups is that the definition cannot be formalized in terms of [[first-order logic]]. This is because doing so would require an axiom of the form |
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which contains an infinite [[Logical disjunction|disjunction]] and is therefore inadmissible: first order logic permits quantifiers over |
which contains an infinite [[Logical disjunction|disjunction]] and is therefore inadmissible: first order logic permits quantifiers over the sort(s) of individuals and cannot capture properties or "subsets" of that sort. It is also not possible to get around this infinite disjunction by using an infinite set of axioms: the [[compactness theorem]] implies that no set of first-order formulae can characterize the periodic groups.{{cite book|last1=Ebbinghaus|first1=H.-D.|last2=Flum|first2=J.|last3=Thomas|first3=W.|title=Mathematical logic|year=1994|publisher=Springer|location=New York [u.a.]|isbn=978-0-387-94258-2|pages=[https://archive.org/details/mathematicallogi1996ebbi/page/50 50]|url=https://archive.org/details/mathematicallogi1996ebbi/page/50|edition=2. ed., 4. pr.|accessdate=18 July 2012|quote=However, in first-order logic we may not form infinitely long disjunctions. Indeed, we shall later show that there is no set of first-order formulas whose models are precisely the periodic groups.}} |
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== Related notions == |
== Related notions == |
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