The Berry Paradox

The Berry paradox is given as:

‘The smallest natural number that cannot be defined by seventeen words or by less than seventeen words’

Analyzing this, we can see that it is an attempt to define something by:

1. defining it semantically, by considering what expressions constitute the definition of a natural number according to the syntax of the given language,
   and at the same time,
2. defining it with no consideration of the semantics, instead defining it by the physical properties of the sequence of symbols.

In the first case (a), we are defining an expression in the normal way within the language, defining it by the semantic meaning of the expression according to the rules of syntax of the language. In such a definition, we can observe that one can have multiple different expressions that have the same meaning in that language, but which are expressions of different combinations of symbols. Clearly there can be many such expressions which have different lengths of sequences of symbols, or which contain different numbers of words, but which are all equivalent expressions in that language. And any expression can be substituted for any other equivalent expression without altering the meaning in that language.

In the second case (b), however, we are defining an expression in meta-mathematical terms, by defining an expression purely in terms of the physical properties of the sequence of symbols that make up that expression, and ignoring any meaning whatsoever of the expression. This can be done in several ways; in this case it is by the number of words that constitute the expression.

In a natural language. such as English, this is a case of the language acting both as the language itself, but at the same time also as a meta-language to itself. And so the expression attempts to be, at the same time, an object in English, and also valid syntax of English. This means that the expression is not a logically valid proposition.

In a well-defined formal system, this cannot happen, but in natural language, it can. It is pointless asking how to create a “solution” to the paradox for natural language, since in a natural language such as English, such paradoxes are always possible, since such paradoxes arise from the ambiguity of natural language that allows expressions to be at the same time, objects of the language, and valid syntax of the language. This allows what are called the self-referential paradoxes - paradoxes where an expression appears to refer to itself; see also the Liar Paradox for more on this.

Other paradoxes