a nontrivial language property

In Rice's theorem the word 'non-trivial' is doing heavy lifting, and it has a precise, narrow meaning worth pinning down. A property of languages is a yes/no test you apply to a set of strings: 'is it empty?', 'does it contain hello?', 'is it finite?'. Such a property is non-trivial when it is not the same answer for every recognizable language — that is, at least one recognizable language passes the test and at least one fails it. Trivial means the opposite: the test gives the same verdict to every language, so it carries no information.

There are exactly two trivial properties, and they are both easily decidable, which is why Rice's theorem must exclude them. One is the always-true property: 'L(M) is a recognizable language' is true for every machine, so a decider can just print yes. The other is the always-false property: 'L(M) is not a recognizable language' is false for every machine M (its language is recognizable by definition of M), so a decider prints no. Neither tells you anything about a specific machine, and both are decidable precisely because the answer never depends on M.

Every other property — the ones that actually distinguish some machines from others — is non-trivial, and Rice's theorem makes all of them undecidable. Emptiness is non-trivial because some machines recognize the empty language and others do not. Finiteness, regularity, containing a particular string: all non-trivial, all undecidable. The takeaway is sharp: the moment a behavioural property is interesting enough to separate one program's behaviour from another's, no algorithm can decide it in general.