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## Friday, July 30, 2010

### Words and Numbers...

The Berry Paradox comes in a few different forms. Here is one of the simplest examples of it for easy comprehension:

Name "the smallest possible integer NOT definable by fewer than twelve words".

It's easy to imagine examples (definitions) that DON'T work:

the number of toes on your right foot [only 8 words, to define the number "5"]
the number of inches in a foot [7 words to define the number "12"]
the number of seconds in a million years [8 words to define whatever the number would be]

easy enough...

The problem arises however (if it isn't already obvious), that if you did somehow find an integer that you could only define with a sentence of 12 or more words, and it WAS the smallest such definable integer, THEN, IT could be accurately designated (defined) by the original 11-word sentence above ("the smallest possible integer not definable by fewer than twelve words") --- thus a self-referential contradiction!

...Another example of where mixing language/semantics with numbers/mathematics proves vexing, throwing light on illogical ambiguity or paradox within language. Much of the history of underlying problems with mathematical consistency entails issues of self-reference.

More on the Berry paradox via Wikipedia: