Why does the fixed point theorem hold for every lambda term?

Question

Can someone give a clear and simple answer for why the fixed point theorem holds for every $\lambda$-term, in contrast with the fact that not all numerical function have a fixed point?

Answer 1

As was noted by others lambda terms are more general than numerical functions so you are able to draw answer(fixed point) from different set. While for numerical function e.g. $f: \mathbb{R} \rightarrow \mathbb{R}$ your fixed point must belong to $\mathbb{R}$. On contrary lambda terms reduces to other lambda terms some of them can designate numbers from $\mathbb{R}$ some of them do not designate the numbers.

For more rigorous explanations you should read some proofs of fixed point theorem for lambda calculus.