What is the definition of a property?

Question

I have seen 2 answers in stackoverflow:

1. A "trivial" property is one that holds either for all languages or for none.

2. The property is trivial if it contains every TM, or if it is empty.

My problem is: Saying that a property is trivial if it contains every TM is not the same as saying that a property is trivial if it contain all the languages (including non RE languages).

Same as for: Saying that a property is trivial if it is does not contain any TM languages is not the same as saying that a property is trivial if it is empty.

Answer 1

We call a set of languages, $P\subseteq 2^{\Sigma^*}$, a property. If you think of this subset as the set of languages who satisfy some property, then we can simply say that a language $L$ satisfies the property iff $L\in P$.

Rice theorem tells you that you can't, given a Turing machine $M$, check if $L(M)$ satisfies some non trivial property $P$, i.e. $P\neq \emptyset$, $P\nsupseteq RE$. Note that Shaull makes a distinction between semantic and syntactic properties in the comments. To the best of my knowledge, the standard definition of a property in the context of Rice's theorem is a set of languages.