C-minimal theory: Difference between revisions

Revision as of 01:37, 15 February 2022

Revision as of 01:38, 15 February 2022

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==Definition==

A ''C''-relation is a ternary relation ''C''(''x'';''yzy,z'') that satisfies the following axioms.

# <math>\forall xyz\, [ C(x;yzy,z)\rightarrow C(x;zyz,y) ],</math>

# <math>\forall xyz\, [ C(x;yzy,z)\rightarrow\neg C(y;xzx,z) ],</math>

# <math>\forall xyzw\, [ C(x;yzy,z)\rightarrow (C(w;yzy,z)\vee C(x;wzw,z)) ],</math>

# <math>\forall xy\, [ x\neq y \rightarrow \exists z\neq y\, C(x;yzy,z) ].</math>

A '''C-minimal structure''' is a [[structure (mathematical logic)|structure]] ''M'', in a [[signature (logic)|signature]] containing the symbol ''C'', such that ''C'' satisfies the above axioms and every set of elements of ''M'' that is definable with parameters in ''M'' is a Boolean combination of instances of ''C'', i.e. of formulas of the form ''C''(''x'';''bc''), where ''b'' and ''c'' are elements of ''M''.