- Axiom 3.7. (Infinity). There exists a set \(\mathbb{N}\), whose elements are called natural numbers, as well as an object \(0\) in \(\mathbb{N}\), and an object \(n\pp\) assigned to every natural number \(n\in\mathbb{N}\), such that the Peano axioms (Axioms 2.1-2.5) hold.
Axiom 3.7
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Exercise 3.2.1
Show that the universal specification axiom, Axiom 3.8, if assumed to be true, would imply Axioms 3.2, 3.3, 3.4, 3.5, and 3.6. (If we assume that all natrual numbers are object, we also obtain Axiom 3.7.) Thus, this axiom, if permitted, would simplify the foundations of set theory tremendously (and can be viewed as one basis for an intuitive model of set theory known as "naive set theory"). Unfortunately, as we have seen, Axiom 3.8 is "too good to be true"!
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Chapter3
Exercise 3.2.1. Show that the universal specification axiom, Axiom 3.8, if assumed to be true, would imply Axioms 3.2, 3.3, 3.4, 3.5, and 3.6. (If we assume that all natrual numbers are object, we also obtain Axiom 3.7.) Thus, this axiom, if permitted, would simplify the foundations of set theory tremendously (and can be viewed as one basis for an intuitive model of set theory known as "naive set theory"). Unfortunately, as we have seen, Axiom 3.8 is "too good to be true"!