  A rather easy yet rigorous proof of a version of G\"odel's first
incompleteness theorem is presented. The version is "each recursively
enumerable theory of natural numbers with 0, 1, +, *, =, logical and, logical
not, and the universal quantifier either proves a false sentence or fails to
prove a true sentence". The proof proceeds by first showing a similar result on
theories of finite character strings, and then transporting it to natural
numbers, by using them to model strings and their concatenation. Proof systems
are expressed via Turing machines that halt if and only if their input string
is a theorem. This approach makes it possible to present all but one parts of
the proof rather briefly with simple and straightforward constructions. The
details require some care, but do not require significant background knowledge.
The missing part is the widely known fact that Turing machines can perform
complicated computational tasks.
