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The Axiom of Infinity and the Natural Numbers Axiom of Infinity Natural Numbers Axiomatic Systems The Axiom of Infinity and The Natural Numbers Bernd Schroder¨ logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers 1. The axioms that we have introduced so far provide for a rich theory. 2. But they do not guarantee the existence of infinite sets. 3. In fact, the superstructure over the empty set is a model that satisfies all the axioms so far and which does not contain any infinite sets. (Remember that the superstructure itself is not a set in the model.) Axiom of Infinity Natural Numbers Axiomatic Systems Infinite Sets logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers 2. But they do not guarantee the existence of infinite sets. 3. In fact, the superstructure over the empty set is a model that satisfies all the axioms so far and which does not contain any infinite sets. (Remember that the superstructure itself is not a set in the model.) Axiom of Infinity Natural Numbers Axiomatic Systems Infinite Sets 1. The axioms that we have introduced so far provide for a rich theory. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers 3. In fact, the superstructure over the empty set is a model that satisfies all the axioms so far and which does not contain any infinite sets. (Remember that the superstructure itself is not a set in the model.) Axiom of Infinity Natural Numbers Axiomatic Systems Infinite Sets 1. The axioms that we have introduced so far provide for a rich theory. 2. But they do not guarantee the existence of infinite sets. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers (Remember that the superstructure itself is not a set in the model.) Axiom of Infinity Natural Numbers Axiomatic Systems Infinite Sets 1. The axioms that we have introduced so far provide for a rich theory. 2. But they do not guarantee the existence of infinite sets. 3. In fact, the superstructure over the empty set is a model that satisfies all the axioms so far and which does not contain any infinite sets. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers Axiom of Infinity Natural Numbers Axiomatic Systems Infinite Sets 1. The axioms that we have introduced so far provide for a rich theory. 2. But they do not guarantee the existence of infinite sets. 3. In fact, the superstructure over the empty set is a model that satisfies all the axioms so far and which does not contain any infinite sets. (Remember that the superstructure itself is not a set in the model.) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers There is a set I that contains /0 as an element, and for each a 2 I the set a [ fag is also in I. In some ways this axiom says we can “cut across” the different levels of a superstructure and still obtain a set. The superstructure over I is a model that satisfies all axioms introduced so far. Axiom of Infinity Natural Numbers Axiomatic Systems The Axiom of Infinity logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers In some ways this axiom says we can “cut across” the different levels of a superstructure and still obtain a set. The superstructure over I is a model that satisfies all axioms introduced so far. Axiom of Infinity Natural Numbers Axiomatic Systems The Axiom of Infinity There is a set I that contains /0 as an element, and for each a 2 I the set a [ fag is also in I. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers The superstructure over I is a model that satisfies all axioms introduced so far. Axiom of Infinity Natural Numbers Axiomatic Systems The Axiom of Infinity There is a set I that contains /0 as an element, and for each a 2 I the set a [ fag is also in I. In some ways this axiom says we can “cut across” the different levels of a superstructure and still obtain a set. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers Axiom of Infinity Natural Numbers Axiomatic Systems The Axiom of Infinity There is a set I that contains /0 as an element, and for each a 2 I the set a [ fag is also in I. In some ways this axiom says we can “cut across” the different levels of a superstructure and still obtain a set. The superstructure over I is a model that satisfies all axioms introduced so far. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers (Existence of the natural numbers.) There is a set, denoted N and called the set of natural numbers, so that the following hold. 1. There is a special element in N, which we denote by 1. 0 2. For each n 2 N, there is a corresponding element n 2 N, called the successor of n. 3. The element 1 is not the successor of any natural number. 4. Principle of Induction. If S ⊆ N is such that 1 2 S and for 0 each n 2 S we also have n 2 S, then S = N. 0 0 5. For all m;n 2 N if m = n , then m = n. The above properties are also called the Peano Axioms for the natural numbers. Axiom of Infinity Natural Numbers Axiomatic Systems Theorem. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers There is a set, denoted N and called the set of natural numbers, so that the following hold. 1. There is a special element in N, which we denote by 1. 0 2. For each n 2 N, there is a corresponding element n 2 N, called the successor of n. 3. The element 1 is not the successor of any natural number. 4. Principle of Induction. If S ⊆ N is such that 1 2 S and for 0 each n 2 S we also have n 2 S, then S = N. 0 0 5. For all m;n 2 N if m = n , then m = n. The above properties are also called the Peano Axioms for the natural numbers. Axiom of Infinity Natural Numbers Axiomatic Systems Theorem. (Existence of the natural numbers.) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers 1. There is a special element in N, which we denote by 1. 0 2. For each n 2 N, there is a corresponding element n 2 N, called the successor of n. 3. The element 1 is not the successor of any natural number. 4. Principle of Induction. If S ⊆ N is such that 1 2 S and for 0 each n 2 S we also have n 2 S, then S = N. 0 0 5. For all m;n 2 N if m = n , then m = n. The above properties are also called the Peano Axioms for the natural numbers. Axiom of Infinity Natural Numbers Axiomatic Systems Theorem. (Existence of the natural numbers.) There is a set, denoted N and called the set of natural numbers, so that the following hold. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers 0 2. For each n 2 N, there is a corresponding element n 2 N, called the successor of n. 3. The element 1 is not the successor of any natural number. 4. Principle of Induction. If S ⊆ N is such that 1 2 S and for 0 each n 2 S we also have n 2 S, then S = N. 0 0 5. For all m;n 2 N if m = n , then m = n. The above properties are also called the Peano Axioms for the natural numbers. Axiom of Infinity Natural Numbers Axiomatic Systems Theorem. (Existence of the natural numbers.) There is a set, denoted N and called the set of natural numbers, so that the following hold. 1. There is a special element in N, which we denote by 1. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers 3. The element 1 is not the successor of any natural number. 4. Principle of Induction. If S ⊆ N is such that 1 2 S and for 0 each n 2 S we also have n 2 S, then S = N. 0 0 5. For all m;n 2 N if m = n , then m = n. The above properties are also called the Peano Axioms for the natural numbers. Axiom of Infinity Natural Numbers Axiomatic Systems Theorem. (Existence of the natural numbers.) There is a set, denoted N and called the set of natural numbers, so that the following hold. 1. There is a special element in N, which we denote by 1. 0 2. For each n 2 N, there is a corresponding element n 2 N, called the successor of n. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers 4.
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