Logical-principle and simple formal-system theory {metamathematics} {proof theory, mathematics} can have no infinities, use minimum English, use existence theorems to show how to construct new objects, not use proof by contradiction, not use Zorn's lemma, and not use axiom of choice [Hilbert, 1899] [Kleene, 1952] [Tarski, 1983].
Axiomatic-theory object languages can use higher-level natural-language terms {metalanguage} {syntax language}. Higher-level languages express theorems {metatheorem} about proofs, formal theories, languages, and logic [Hilbert, 1899] [Kleene, 1952] [Tarski, 1983]. Metamathematics is a metalanguage.
Mathematics can use only ideas of relation and class {logic of relations} {pure mathematics}.
3-Mathematics-Axiomatic Theory
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Date Modified: 2022.0225