PUMaC 2025 · 加试 · 第 1 题

Problem 1.3.2 (5 points) Let F be a class function, and suppose that dom ( F ) is a set. Prove that ran ( F ) is a set, and conclude that F is also a set. 1.4 Philosophical Discussion: The Meta Theory In the previous problems, you used the axioms of ZFC to prove many statements – that is, sentences. However, if you look carefully, you might notice that some of the problem statements aren’t sentences. Notably, Problem 1.3.2 starts by picking an arbitrary class function F , which is not allowed in a sentence (as classes are not sets). So, what did you actually do by solving the problem? 1 Recall that the axioms of separation and replacement are actually axiom schemata: they consist of infinitely many axioms. Similarly, we may think of solving Problem 1.3.2 as proving infinitely many sentences at once: for every class F , you prove the sentence that if F is a class function and dom( F ) is a set, then ran( F ) and F are sets. That is, the statement of Problem 1.3.2 is a meta-mathematical statement, instead of a mathematical statement (i.e. a formula). To clarify this distinction, we introduce the terms base theory and meta theory . Sets live in the base theory, and when we prove sentences, we are working in the base theory. In contrast, meta-mathematical objects, like formulas or classes, live in the meta theory, and reasoning about them constitutes working in the meta theory. For most problems in this Power Round, the distinction between the base theory and the meta theory can be mostly handwaved away. However, if you are not careful, you might still make mistakes! It is especially important to keep this in mind in the later sections, as there will be a blend of mathematical and meta-mathematical concepts. Finally, if you are worried about the abuse of terminology where we say that certain classes “are” sets, rest assured that this will not cause any problems. Every time such an abuse of terminology occurs, it is always possible to rewrite things such that the abuse does not occur, often with the expense of making everything more cumbersome. 1 The plural form of “schema”. 11 2 Ordinals (180 points) Ordinals are one of the most important concepts in set theory. Intuitively, they give us a 2 way of counting past infinity. The natural numbers 0 , 1 , 2 , . . . are ordinals, but beyond that, we have the ordinal ω , the smallest infinite ordinal. After that, we have ω + 1, then ω + 2, and so on, then ω + ω = ω · 2, and on and on and on... As sets, each ordinal α is, intuitively, the set of ordinals smaller than α . For instance, since there are no ordinals less than 0, we have 0 = ∅ . Next, we have 1 = { 0 } = { ∅ } , and 2 = { 0 , 1 } = { ∅ , { ∅ }} , and then ω = { 0 , 1 , 2 , . . . } , and ω + 1 = { 0 , 1 , 2 , . . . , ω } , and so on and so forth. This informal idea will be made rigorous below. 2.1 The Basics of Ordinals (75 points) Definition 2.1.1 — A class x is transitive if any element of x is a subclass of x . In other words, if x is transitive, then z ∈ y and y ∈ x imply z ∈ x . For example, the sets ∅ and { ∅ , { ∅ } , {{ ∅ }}} are transitive, but {{ ∅ }} is not transitive. Definition 2.1.2 — A set α is an ordinal if α is transitive, and every element of α is also transitive. The class of ordinals is denoted Ord. For example, ∅ and { ∅ , { ∅ }} are ordinals, but { ∅ , { ∅ } , {{ ∅ }}} (a transitive set) is not an ordinal, because it contains an element {{ ∅ }} which is not transitive.