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Lo¨wenheim-Skolem and the real numbers

16 March, 2015 - 11:47
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The Lo¨wenheim-Skolem theorem of logic states that if a set of (countable) domain axioms has a model at all, then it has a countable model. This is a bit surprising when applied to the axioms of arithmetic for the real numbers: even though the real numbers are uncountable, there is some countable model which meets all our (fnite) axioms of the real numbersl