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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 https://www.macmillandictionary.com/dictionary/british/assert 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.

Robert
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  • That link says prove and show and point to are associated words for assert, but they need not be synonyms. Here they are not synonyms for "state firmly that something is true" – Henry Feb 15 '20 at 16:39
  • @Henry https://www.macmillanthesaurus.com/assert "Synonyms: To show or agree that something is true" – Robert Feb 15 '20 at 17:11

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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.

Noah Schweber
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