# Undecidable vs Unsolvable?

In decidability theory, I understand that if a problem is labeled "decidable", then we can construct a Turing Machine that definitively tells us whether an input is valid or invalid. My question is what does "undecidable" mean ? I read the "undecidable" applies to problems that have Turing Machines but cannot definitively tell us whether an input is valid or not(i.e. it outputs "yes" for valid inputs, and keep going for others indefinitely such that we will not know when, if ever, it will output anything). Then does "undecidable" applies to problems that can not be represented by Turing Machines at all, also? What is the difference between unsolvable and undecidable ?

EDIT: I was reading my textbook(Introduction to Automata Theory, Hopcroft) and I ran into this: "Remember that showing there is no Turing machine at all for a language is showing something stronger than that the language is undecidable(i.e., that it has no algorithm, or TM that always halts"(pg 388). This seems to imply undecidable problems do not include ones without any TMs. Am I interpreting this correctly ?

• Mathematically "unsolvable" usually means an equation which has no analytical solution for example 0=1+1/(2^x)+1/(3^x) – Niklas Rosencrantz May 22 '16 at 18:00
• "I understand that if a problem is labeled "decidable", then we can construct a Turing Machine" -- well, no. "there exists" != "we can construct"; see here. – Raphael May 25 '16 at 0:21
• @Raphael I guess I didn't know that(thanks for pointing it out). Any ideas on my main question though ? It appears(to me) the textbook is saying undecidable does not include those problems without any Turing machines but answers here indicate otherwise. – Jenna Maiz May 25 '16 at 1:42