TITLE: Is there an example for an undefinable number?
QUESTION [15 upvotes]: This question is motivated by a comment of Robert on the question Can any Real number be typed in a computer? :

Can you "think of" an undefinable number? – Robert Israel

I would like to reformulate this (ironic) question to:

Can you give an (concrete) example for an undefinable number?

My first guess would be "No!" because when you give an example of a number you need a definition for it which contradicts that the number shall be undefinable. But we are talking about mathematics and I learned that the first guess is not always true ;-) So I can imagine that finally the answer is "Yes" to the above question (maybe it is possible to give an undefinable number in first order logic by using higher order logic).
Note: I mean "undefinable" in the sense "undefinable in first order logic".

REPLY [8 votes]: If the choice of language is the type of language we use in everyday life, Ian Stewart's How to Cut a Cake gives the example of Richard's paradox:

"In the English language, some sentences define positive integers and others do not. For example 'The year of the Declaration of Independence' defines the number 1776, whereas 'The historical significance of the Declaration of Independence' does not define a number. So what about this sentence: 'The smallest number that cannot be defined by a sentence in the English language containing fewer than twenty words.' Observe that whatever this number may be, we have just defined it using a sentence in the English language containing only nineteen words. Oops."

Of course, this can also be extended to Spanish, French, German, etc., instead of just English.