21. Additional examples of inconsistent theories arise from the axiom schema of unrestricted comprehension is assumed in set theory. 22. These axioms are all axiom schema of induction makes redundant the axioms 3, 10, and 11. 23. The Hoare axiom for assignment is not given as a single axiom but rather as an axiom schema . 24. The next four are general statements about second-order induction axiom with a first-order axiom schema . 25. The axiom schema of specification must be used to reduce this to a set with exactly these two elements. 26. Axiomatizing arithmetic induction in first-order logic requires an axiom schema containing a separate axiom for each possible predicate. 27. The axiom schema of collection is equivalent to the axiom schema of replacement over the remainder of the ZF axioms. 28. The axiom schema of collection is equivalent to the axiom schema of replacement over the remainder of the ZF axioms. 29. Thus, one may use this as an axiom schema in the place of the axioms of empty set and pairing. 30. For this reason, the axiom schema of separation is often left out of modern lists of the Zermelo Fraenkel axioms.
