Editing Rice's theorem (section)

==Formal statement==
Let ''φ'' be an [[admissible numbering]] of [[partial computable function]]s. Let ''P'' be a subset of <math>\mathbb{N}</math>. Suppose that:
# ''P'' is ''non-trivial'': ''P'' is neither empty nor <math>\mathbb{N}</math> itself.
# ''P'' is ''extensional'': for all integers ''m'' and ''n'', if ''φ''<sub>''m''</sub> = ''φ''<sub>''n''</sub>, then ''m'' ∈ ''P'' ⟺ ''n'' ∈ ''P''.
Then ''P'' is [[undecidable problem|undecidable]].

A more concise statement can be made in terms of [[index set (computability) | index sets]]: The only decidable index sets are ∅ and <math>\mathbb{N}</math>.