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?

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.