Definable real number: Difference between revisions

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Each set [[Model theory|model]] ''M'' of ZFC set theory that contains uncountably many real numbers must contain real numbers that are not definable within ''M'' (without parameters). This follows from the fact that there are only countably many formulas, and so only countably many elements of ''M'' can be definable over ''M''. Thus, if ''M'' has uncountably many real numbers, we can prove from "outside" ''M'' that not every real number of ''M'' is definable over ''M''.

This argument becomes more problematic if it is applied to class models of ZFC, such as the [[von Neumann universe]].{{cn|date=May 2021}} The argument that applies to set models cannot be directly generalized to class models in ZFC because the property "the real number ''x'' is definable over the class model ''N''" cannot be expressed as a formula of ZFC. Similarly, the question of whether the von Neumann universe contains real numbers that it cannot define cannot be expressed as a sentence in the language of ZFC. Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable.{{sfn|Hamkins|Linetsky|Reitz|2013}}

== See also ==