Morley rank
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In mathematical logic, Morley rank, named after Michael D. Morley, is a means of measuring dimension in model theory via ranks. It was the first such measure and is the one which most closely resembles dimension from algebraic geometry.
For a theory T with model M, a formula φ defining a definable subset S of M has Morley rank at least α (for α a successor ordinal) if in some elementary extension N of M, S has countably many disjoint definable subsets Si, each of rank at least α-1. Note that Morley rank measures the rank of a formula, not a set.
If φ defining S has rank α, and S breaks up into no more than n < ω finite subsets of rank α, then φ is said to have Morley degree n. A formula defining a finite set has Morley rank 0. A formula with Morley rank 1 and Morley degree 1 is called strongly minimal. A strongly minimal structure is one where the trivial formula x=x is strongly minimal. Morley rank and strongly minimal structures are key tools in the proof of Morley's categoricity theorem and in the larger area of stability theory.
[edit] See also
- U-rank
[edit] References
- B. Hart Stability theory and its variants (2000) pp. 131-148 in Model theory, algebra and geometry, edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39, Cambridge Univ. Press, New York, 2000. Contains a formal definition of Morley rank.
- David Marker Model Theory of Differential Fields (2000) pp. 53-63 in Model theory, algebra and geometry, edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39, Cambridge Univ. Press, New York, 2000.
