How does one prove that 1-generic set is not computable?

Question

Without resorting to diagonalization proof of halting problem, how does one prove that 1-generic set is not computable?

Answer 1

The 1-generic set is immune(infinite and contains no infinite c.e. subset), so it's not computable.

We know that 1-generic set is infinite. All we need to do is to show that it doesn's contain any infinite c.e. subset.

Let $A$ be a 1-generic set, then for any c.e. set $W_{e}$ of strings we meet the requirement:

$(\exists \sigma\subset A)[\sigma\in W_{e}\vee(\forall\tau\supseteq\sigma)[\tau\notin W_{e}]]$

Let $B\subseteq A$ be a c.e.set, we define $W_{a}=\{\sigma:\exists x\in B[\sigma(x)=0]\}$. It's easy to see $B$ is finite, otherwise $A$ can't satisfy the requirement for $W_{a}$.