* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Set Theory II
Survey
Document related concepts
Computability theory wikipedia , lookup
Truth-bearer wikipedia , lookup
Model theory wikipedia , lookup
Quantum logic wikipedia , lookup
Laws of Form wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Curry–Howard correspondence wikipedia , lookup
History of the Church–Turing thesis wikipedia , lookup
Law of thought wikipedia , lookup
History of the function concept wikipedia , lookup
Quasi-set theory wikipedia , lookup
Mathematical logic wikipedia , lookup
List of first-order theories wikipedia , lookup
Peano axioms wikipedia , lookup
Principia Mathematica wikipedia , lookup
Transcript
Set Theory II Last time we discussed the Axioms of Extension, Specification, Unordered Pairs, and Unions. Some more aioms of set theory Powers For each set there exists a collection of sets that contains among its elements all the subsets of the given set. (Combined with the Axiom of Specification, it follows that if A is a set, then P(A) = {B : B ⊆ A} is also a set.) Regularity Every non-empty set contains an element that is disjoint from itself. Notes: • The Axiom of Regularity is not in Halmos’ book, and much of set theory can be developed without it, but I am including it here to discuss the issue of a set being an element of itself. • We are omitting the Axiom of Replacement, which says that the image of a set under a function is a set. We will discuss functions in much more details later. 1. (Russell paradox) Prove that “There is no Universe” without using the Axiom of Regularity. (Let A be a set and B = {x ∈ A : x ∈ / x}. Prove that B ∈ / A.) 2. Write the Axiom of Regularity in symbolic logic. 3. Prove that if we assume the Axiom of Regularity (and the four axiom from last class), then a set cannot be an element of itself. (Let A be a set. Apply the Axiom of Regularity to {A}.) 4. Prove that “There is no Universe” using the Axiom of Regularity. 5. Explain why the intersection of an empty collection of sets is undefined. 6. Let A and B be sets. Explain how the Cartesian product A × B can be defined so that its existence follows from the Axioms we have seen.