TITLE: Is there a theorem that says that there is always more than one way to "continue a finite sequence"?
QUESTION [6 upvotes]: I have come across a bit of folklore(?) which goes something like "given any finite sequence of numbers, there is more than one 'valid' way of continuing the sequence". For example see here. I would like to know if this is actually stated and proved rigorously, and if so, where can I find a statement and proof?
EDIT: The first couple of answers reflect my concerns exactly. But to quote from the book review I linked to, 

Wittgenstein's Finite Rule Paradox implies that any finite sequence of numbers can be a continued in a variety of different ways - some natural, others unexpected and surprising but equally valid.

I didn't use the terms "Wittgenstein's Finite Rule Paradox" and "Wittgenstein on rule-following" before as googling them turns up results which look more like philosophy and linguistics than mathematics. My background in logic is nonexistent, I'm looking for any logicians out there who may have seen this before.

REPLY [8 votes]: There is an area of computer science called grammatical inference.
Let $\Sigma$ be a finite alphabet, $\Sigma^*$ be the set of finite length strings over $\Sigma$ and a language be a set of strings. Let $F$ be a family of languages. The data for the problem is a sequence of words $w_0, w_1, \ldots$ from a language in $F$. A learner receives the data and generates a sequence of languages $H_0, H_1, \ldots$ called hypotheses. 
Typically the hypothesis $H_i$ includes all words provided till that point. The learner stabilizes if the hypotheses do not change after a point and is successful if the stable hypothesis is the language from which the words were drawn. A family of languages can be learnt in the limit if there is an algorithm that can successfully learn the source language. 
There is a theorem due to E. Mark Gold (Language Identification in the Limit, 1967) stating that if the family $F$ contains all languages of finite cardinality and at least one language of infinite cardinality, it cannot be learnt in the limit. 
This result may be one formalisation related to the original question, but does not tell the whole story. Dana Angluin (Inference of Reversible Languages, 1982) showed that there are families containing infinitely many languages that can be learnt in the limit. There is a recent text Grammatical Inference: Learning Automata and Grammars by Colin de la Higuera devoted to this area.