Without a prior commitment to a privileged interpretation of arithmetic, 'true but unprovable' collapses into 'unprovable in this system but provable in an extension'—a far weaker claim.

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Arithmetic: Arithmetic is the branch of mathematics dealing with basic number operations—addition, subtraction, multiplication, and division. It's the foundation of math that you use in everyday life, like calculating change at a store, splitting a bill, or figuring out measurements. Essentially, it's the practical math skill everyone learns early in school to work with numbers in simple, straightforward ways.
Privileged interpretation(as what the author claims we shouldn't assume about arithmetic): A special or favored way of understanding something that is treated as uniquely correct or important, often without sufficient justification.
Unprovable in this system(as contrasted with provable in extensions): A statement that cannot be proven using the specific set of rules and assumptions (called axioms) that define a particular mathematical or logical system.
extension(Semantics and philosophy of language): Another term for reference, i.e., the object or set of objects a term picks out

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