I have always had a hard time making sense of infinities.     Example: the language $L = \{ \epsilon , 01, 0011, 000111, 00001111, ... \}$ is not DFA acceptable because a machine capable of accepting it would require an infinite amount of states.  I can draw a DFA with ellipses in the middle to represent it's expansion.  I can prove that: For all strings x in L, using a machine M = mentioned DFA, under some finite expansion, I can accept all strings y in L, such that $|y| \leq |x|$ and such that M has $|x| + 2$ states and such that all strings in compliment(L) are not accepted by M.All of those machines are finite.  The problem that prevents me from making the leap to claim that L is then DFA acceptable, is that I need to prove that there exists a largest string in L, but there is no largest string in L.Anyways, I know this sounds stupid because knowing what is proven about convergence and divergence, I should be able to make mathematical sense of this.  The problem is that no matter how much I know about infinity, I still do not think that what is said about convergence and divergence seems logical.  I always have this feeling that something is wrong.  So I guess I just wonder how I can look at it so that it makes more sense. In short, when a DFA needs a state for each character of a string in order to accept the string, and the language is an infinite set of strings whose enumeration could be represented as a sequence of strings increasing in length, how is it that a DFA needs an infinite amount of states to accept all of those strings, given each string is finite?
Help me resolve my doubts about infinities, exemplified by an example of non DFA acceptable language
automata;finite automata