Editing Berry paradox (section)

== Resolution ==
The Berry paradox as formulated above arises because of systematic [[ambiguity]] in the word "definable". In other formulations of the Berry paradox, such as one that instead reads: "...not [[name]]able in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to [[vicious circle principle|vicious circle]] fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.{{sfn|Russell|Whitehead|1927}} To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them.

This family of paradoxes can be resolved by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. "The number not nameable<sub>0</sub> in less than eleven words" may be nameable<sub>1</sub> in less than eleven words under this scheme.{{sfn|Quine|1976|p=10}}

However, one can read [[Alfred Tarski]]'s contributions to the [[Liar Paradox]] to find how this resolution in languages falls short. Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

However, this system is incomplete. One would like to be able to make statements such as "For every statement in level ''α'' of the hierarchy, there is a statement at level ''α''+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible).{{sfn|Kripke|1975}}<ref name=PlatoProject.LiarParadox.TarskiHier>{{harvnb|Beall|Glanzberg|Ripley|2016}}</ref> [[Saul Kripke]] is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth,"<ref name=PlatoProject.LiarParadox.TarskiHier/> and it is recognized as a general problem in hierarchical languages.{{sfn|Glanzberg|2015}}<ref name=PlatoProject.LiarParadox.TarskiHier/>