Not Every Decision Problem is decidable

A decision problem $\mathcal{P}=⟨D,Y⟩$ with a language $L[\mathcal{P},\mathrm{enc}]$ over a finite alphabet $\Sigma$ can be identified with a function $f:{\Sigma }^{\ast }\to \{0,1\}$, where $0$ means NO and 1 means YES:

$$f:w\mapsto \{\begin{array}{ll}1& \text{if }w\in Y\\ 0& \text{otherwise}\end{array}$$

Since ${\Sigma }^{\ast }$ is countable, the set of all functions ${\Sigma }^{\ast }\to \{0,1\}$ is uncountable. Every algorithm has a finite description and is therefore countable. Hence, there are strictly more decision problems than algorithms, and most decision problems are not decidable.