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W2004277622 abstract "A function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper V> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined on the set of all subsets of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding=application/x-tex>omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that for each set <italic>K</italic>, the value, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper V Subscript upper K> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {V}_K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is the set of formulas valid in all structures of cardinality in <italic>K</italic>. An analysis is made of the dependence of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper V Subscript upper K> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {V}_K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <italic>K</italic>, For any set <bold>K</bold>, let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis upper K right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {d}}(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Kleene-Post degree to which <italic>K</italic> belongs. It is easily seen that for all infinite sets <italic>K</italic>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis upper K right-parenthesis logical-or 1 less-than-or-equal-to d left-parenthesis script upper V Subscript upper K Baseline right-parenthesis less-than-or-equal-to d left-parenthesis upper K right-parenthesis prime> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∨</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {d}}(K) vee 1 leq d({mathcal {V}_K}) leq {text {d}}(K)’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. On the other hand, we prove that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis script upper V Subscript upper K logical-or upper J Baseline right-parenthesis equals d left-parenthesis script upper V Subscript upper K Baseline right-parenthesis logical-or d left-parenthesis script upper V Subscript upper J Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>K</mml:mi> <mml:mo>∨</mml:mo> <mml:mi>J</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∨</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:mi>J</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {d}}({mathcal {V}_{K vee J}}) = {text {d}}({mathcal {V}_K}) vee {text {d}}({mathcal {V}_J})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and use this to prove that, for any two degrees a and b, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a greater-than-or-equal-to 1 comma a less-than-or-equal-to b less-than-or-equal-to a prime> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>a</mml:mtext> </mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>a</mml:mtext> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>b</mml:mtext> </mml:mrow> <mml:mo>≤</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>a</mml:mtext> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {a}} geq 1,{text {a}} leq {text {b}} leq {text {a}}’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and b r.e. a, there exists a set <italic>K</italic> so that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis upper K right-parenthesis equals a> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>a</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {d}}(K) = {text {a}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis script upper V Subscript upper K Baseline right-parenthesis equals b> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>d</mml:mtext> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>V</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {d}}({mathcal {V}_K}) = b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Various similar results are also included." @default.
W2004277622 created "2016-06-24" @default.
W2004277622 creator A5051370311 @default.
W2004277622 date "1973-01-01" @default.
W2004277622 modified "2023-10-16" @default.
W2004277622 title "Sets of formulas valid in finite structures" @default.
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