This chapter focuses on arithmetical comprehension. Arithmetical comprehension is the most obvious set existence axiom to use when developing analysis in a system based on Peano arithmetic (PA) with set variables. This axiom asserts the existence of a set X of natural numbers for each property φ definable in the language of PA. More precisely, if φ(n) is a property defined in the language of PA plus set variables, but with no set quantifiers, then there is a set X whose members are the natural numbers n such that φ(n). Since all such formulas φ are asserted for, the arithmetical comprehension axiom is really an axiom schema. The reason set variables are allowed in φ is to enable sets to be defined in terms of “given” sets. The reason set quantifiers are disallowed in φ is to avoid definitions in which a set is defined in terms of all sets of natural numbers (and hence in terms of itself). The system consisting of PA plus arithmetical comprehension is called ACA0. This system lies at a remarkable “sweet spot” among axiom systems for analysis.
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