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On random models of finite power and monadic logic

✍ Scribed by Matt Kaufmann; Saharon Shelah

Publisher
Elsevier Science
Year
1985
Tongue
English
Weight
619 KB
Volume
54
Category
Article
ISSN
0012-365X

No coin nor oath required. For personal study only.

✦ Synopsis

For any property d~ of a model (or graph), let ~n(&) be the fraction of models of power n which satisfy &, and let ~(d~) = lim~__~ Izn(d)) if this limit exists. For first-order properties &, it is known that ~(&) must be 0 or 1. We answer a question of K. Compton by proving in a strong way that this 0-1 law can fail if we allow monadic quantification (that is, quantification over sets) in defining the sentence &. In fact, by producing a monadic sentence which codes arithmetic on n with probability ~ = 1, we show that every recursive real is ~(&) for some monadic d).

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