# Rice's theorem applicable to the following language?

Let $$L= \{\langle M \rangle \mid M \text{ halts on } \langle M \rangle \}$$ be a language where $$\langle M \rangle$$ is the Code of the TM $$M$$. $$L$$ is undecidable.

I've heard that I can't use Rice's theorem to proof its undecidability.

But why? I can construct a set $$S = \{f_M \mid f_M(\langle M \rangle)\in \{0,1\}\}$$.

It's clear that $$S$$ is not empty and $$S$$ contains not every TM.

• What are $f_m$ and $f_M$? (I assume one of them is a typo, but what does the other one mean?) – David Richerby Jan 25 '19 at 16:53
• both should be f_M. It describes the function of the Turing Machine M. 1 and 0 stands for accept or reject. For example the TM M' that accepts everything has the function f_M'(x) = 1 – Marc Jan 25 '19 at 17:01

Because there may be multiple $$M$$'s corresponding to the same $$f_M$$. That is to say, you cannot deduce $$\langle M\rangle$$ from $$f_M$$, so you cannot describe $$S$$ in this way.