Is there an undecidable finite language of finite words?

Problem Detail: 

Is there a need for $L\subseteq \Sigma^*$ to be infinite to be undecidable?

I mean what if we choose a language $L'$ be a bounded finite version of $L\subseteq \Sigma^*$, that is $|L'|\leq N$, ($N \in \mathbb{N}$), with $L' \subset L$. Is it possible for $L'$ to be an undecidable language?

I see that there is a problem of "How to choose the $N$ words that $\in$ $L' "$ for which we have to establish a rule for choosing which would be the first $N$ elements of $L'$, a kind of "finite" Kleene star operation. The aim is to find undecidability language without needing an infinite set, but I can't see it.

EDIT Note:

Although I chose an answer, many answers and all comments are important.

Asked By : Hernan_eche

Answered By : Ran G.

Yes, there is a need for $L$ to be infinite in order to be undecidable.

To add up on the answers of Raphael and Sam, you should think about "decidable" as things that a computer-program can solve. The program required is very simple, it just needs to output "Yes" for elements in $L$, or otherwise, say no.

So the more "complex" $L$ is, the longer the program you are required to write. In other words, the longer the program you run, you can check more things... So if someone gives a language $L$ which is finite, say $L=\{ a_1, a_2, \ldots, a_n\}$, you can write the following program:

if INPUT = $a_1$ output Yes; if INPUT = $a_2$ output Yes; ... if INPUT = $a_n$ output Yes; output No; 

Now, if some one gives you a larger $L$ (yet finite), you will just write a longer program. This is always true, and any finite $L$ will have it's own program. The only "interesting" case is what happens when $L$ is infinite - your program cannot be infinite.

The issue of "undecidability" is even more interesting: its those (infinite) $L$'s that have no program that works correctly for them. We know that such languages must exists since there are way more (infinite) languages $L$ than the number of programs of finite (but unbounded) length.

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Question Source : http://cs.stackexchange.com/questions/1990