I'm reading a book, and this book says "if no two asserted statements of this theory contradict each other" this got me super confused for some reason, and so I asked this question Consistency definition and now I understand that "asserted statements" mean "all the statements that are provable" so I googled "asserted" and here it says that assert and prove are synonyms while in other dictionaries (Merriam Webster, Cambridge etc) it does not.
P.S. I'm not sure if I should have asked this question on math StackExchange or the one for languages, but since we're talking about math maybe it has a slightly different meaning.
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$\begingroup$In natural language, "asserts" and "proves" mean very different things; an assertion need not come with a justification, while a proof is a justification. In this specific context, however, "asserts" is being used synonymously with "proves:" when the author says that a theory asserts something, they mean that the theory proves that thing. This can be (and has been here) confusing to someone new to the topic, and in my opinion was a bad linguistic choice here.
It's worth noting that sometimes "thinks" is also used this way (e.g. "The theory $T$ thinks that $*$ is commutative" is slang for "$T$ proves $\forall x,y(x*y=y*x)$"), as is "says." This is all part of a general rhetorical device of anthropomorphizing theories which after some initial experience this language can actually be helpful (although of course this is subjective), but I would say it should be avoided early on.
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