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Theories of arithmetic in first-order logic, such as Peano arithmetic, second-order arithmetic, Heyting arithmetic, and their subsystems and extensions. Models of Peano arithmetic.
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vote
What is induction up to $\varepsilon_0$?
Maybe it's spelled out in a more convenient way in Wikipedia here
(about Goodstein sequences), or in the page about Gentzen's consistency proof of Peano's arithmetic.
Hope this helps.