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

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 ∈ I the set a ∪ {a} 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 ∈ I the set a ∪ {a} 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 ∈ I the set a ∪ {a} 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 ∈ I the set a ∪ {a} 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 ∈ N, there is a corresponding element n ∈ 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 ∈ S and for 0 each n ∈ S we also have n ∈ S, then S = N. 0 0 5. For all m,n ∈ 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 ∈ N, there is a corresponding element n ∈ 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 ∈ S and for 0 each n ∈ S we also have n ∈ S, then S = N. 0 0 5. For all m,n ∈ 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 ∈ N, there is a corresponding element n ∈ 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 ∈ S and for 0 each n ∈ S we also have n ∈ S, then S = N. 0 0 5. For all m,n ∈ 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 ∈ N, there is a corresponding element n ∈ 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 ∈ S and for 0 each n ∈ S we also have n ∈ S, then S = N. 0 0 5. For all m,n ∈ 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 ∈ S and for 0 each n ∈ S we also have n ∈ S, then S = N. 0 0 5. For all m,n ∈ 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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 4. Principle of Induction. If S ⊆ N is such that 1 ∈ S and for 0 each n ∈ S we also have n ∈ S, then S = N. 0 0 5. For all m,n ∈ 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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 0 0 5. For all m,n ∈ 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 ∈ N, there is a corresponding element n ∈ 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 ∈ S and for 0 each n ∈ S we also have n ∈ S, then S = N.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers The above properties are also called the Peano Axioms for the natural numbers.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers (Defining N.) Let I be the set from the Axiom of Infinity. Let 1 := {/0} = /0 ∪ {/0} ∈ I. For each n ∈ I, let n0 := n ∪ {n}. Call a subset S ⊆ I a successor set iff /0 6∈ S, 1 ∈ S and for all n ∈ S we have that n0 ∈ S. Then I \{/0} is a successor set. \ Moreover, all successor sets are subsets of I. Define N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers Let I be the set from the Axiom of Infinity. Let 1 := {/0} = /0 ∪ {/0} ∈ I. For each n ∈ I, let n0 := n ∪ {n}. Call a subset S ⊆ I a successor set iff /0 6∈ S, 1 ∈ S and for all n ∈ S we have that n0 ∈ S. Then I \{/0} is a successor set. \ Moreover, all successor sets are subsets of I. Define N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof. (Deﬁning N.)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Let 1 := {/0} = /0 ∪ {/0} ∈ I. For each n ∈ I, let n0 := n ∪ {n}. Call a subset S ⊆ I a successor set iff /0 6∈ S, 1 ∈ S and for all n ∈ S we have that n0 ∈ S. Then I \{/0} is a successor set. \ Moreover, all successor sets are subsets of I. Deﬁne N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof. (Deﬁning N.) Let I be the set from the Axiom of Inﬁnity.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers For each n ∈ I, let n0 := n ∪ {n}. Call a subset S ⊆ I a successor set iff /0 6∈ S, 1 ∈ S and for all n ∈ S we have that n0 ∈ S. Then I \{/0} is a successor set. \ Moreover, all successor sets are subsets of I. Deﬁne N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof. (Deﬁning N.) Let I be the set from the Axiom of Inﬁnity. Let 1 := {/0} = /0 ∪ {/0} ∈ I.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Call a subset S ⊆ I a successor set iff /0 6∈ S, 1 ∈ S and for all n ∈ S we have that n0 ∈ S. Then I \{/0} is a successor set. \ Moreover, all successor sets are subsets of I. Deﬁne N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof. (Deﬁning N.) Let I be the set from the Axiom of Inﬁnity. Let 1 := {/0} = /0 ∪ {/0} ∈ I. For each n ∈ I, let n0 := n ∪ {n}.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Then I \{/0} is a successor set. \ Moreover, all successor sets are subsets of I. Deﬁne N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers \ Moreover, all successor sets are subsets of I. Deﬁne N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers \ Deﬁne N := S to be the intersection of the set S of all successor sets.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof. (Deﬁning N.) Let I be the set from the Axiom of Inﬁnity. Let 1 := {/0} = /0 ∪ {/0} ∈ I. For each n ∈ I, let n0 := n ∪ {n}. Call a subset S ⊆ I a successor set iff /0 6∈ S, 1 ∈ S and for all n ∈ S we have that n0 ∈ S. Then I \{/0} is a successor set. Moreover, all successor sets are subsets of I.

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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers (There is a special element in N, which we denote by 1.) \ Every successor set contains 1. Therefore 1 ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 1.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers \ Every successor set contains 1. Therefore 1 ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 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 \ Therefore 1 ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 1. (There is a special element in N, which we denote by 1.) Every successor set contains 1.

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

Proof of part 1. (There is a special element in N, which we denote by 1.) \ Every successor set contains 1. Therefore 1 ∈ S = N, as was to be proved.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers (For each n ∈ N, there is a corresponding element n0 ∈ , called the successor of n.) N \ Let n ∈ N. Because n ∈ N = S , we conclude that n ∈ S for all S ∈ S . By deﬁnition of successor sets, n0 = n ∪ {n} ∈ S for 0 \ all S ∈ S . Hence n ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 2.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers \ Let n ∈ N. Because n ∈ N = S , we conclude that n ∈ S for all S ∈ S . By deﬁnition of successor sets, n0 = n ∪ {n} ∈ S for 0 \ all S ∈ S . Hence n ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 2. (For each n ∈ N, there is a corresponding 0 element n ∈ 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 \ Because n ∈ N = S , we conclude that n ∈ S for all S ∈ S . By deﬁnition of successor sets, n0 = n ∪ {n} ∈ S for 0 \ all S ∈ S . Hence n ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 2. (For each n ∈ N, there is a corresponding 0 element n ∈ N, called the successor of n.) Let n ∈ N.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers By deﬁnition of successor sets, n0 = n ∪ {n} ∈ S for 0 \ all S ∈ S . Hence n ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 2. (For each n ∈ N, there is a corresponding element n0 ∈ , called the successor of n.) N \ Let n ∈ N. Because n ∈ N = S , we conclude that n ∈ S for all S ∈ S .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 0 \ Hence n ∈ S = N, as was to be proved.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 2. (For each n ∈ N, there is a corresponding element n0 ∈ , called the successor of n.) N \ Let n ∈ N. Because n ∈ N = S , we conclude that n ∈ S for all S ∈ S . By deﬁnition of successor sets, n0 = n ∪ {n} ∈ S for all S ∈ S .

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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers (The element 1 is not the successor of any natural number.) Suppose for a contradiction that 1 was the successor 1 = x0 of 0 an x ∈ N. Then {/0} = 1 = x = x ∪ {x}. This implies x = /0, but /0 6∈ N. We have arrived at a contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 3.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Suppose for a contradiction that 1 was the successor 1 = x0 of 0 an x ∈ N. Then {/0} = 1 = x = x ∪ {x}. This implies x = /0, but /0 6∈ N. We have arrived at a contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 3. (The element 1 is not the successor of any natural number.)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Then {/0} = 1 = x0 = x ∪ {x}. This implies x = /0, but /0 6∈ N. We have arrived at a contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 3. (The element 1 is not the successor of any natural number.) Suppose for a contradiction that 1 was the successor 1 = x0 of an x ∈ N.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers This implies x = /0, but /0 6∈ N. We have arrived at a contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 3. (The element 1 is not the successor of any natural number.) Suppose for a contradiction that 1 was the successor 1 = x0 of 0 an x ∈ N. Then {/0} = 1 = x = x ∪ {x}.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers , but /0 6∈ N. We have arrived at a contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 3. (The element 1 is not the successor of any natural number.) Suppose for a contradiction that 1 was the successor 1 = x0 of 0 an x ∈ N. Then {/0} = 1 = x = x ∪ {x}. This implies x = /0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers We have arrived at a contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers (If S ⊆ N is such that 1 ∈ S and for each n ∈ S 0 we also have n ∈ S, then S = N.) Let S ⊆ N be so that 1 ∈ S and so that for every n ∈ S we have 0 that n ∈ S. Because S is a successor set, by definition of N we conclude N ⊆ S. But by definition of S, we also have S ⊆ N. Hence S = N.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 4.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers Let S ⊆ N be so that 1 ∈ S and so that for every n ∈ S we have 0 that n ∈ S. Because S is a successor set, by definition of N we conclude N ⊆ S. But by definition of S, we also have S ⊆ N. Hence S = N.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 4. (If S ⊆ N is such that 1 ∈ S and for each n ∈ S 0 we also have n ∈ S, then S = N.)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The Natural Numbers Because S is a successor set, by definition of N we conclude N ⊆ S. But by definition of S, we also have S ⊆ N. Hence S = N.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 4. (If S ⊆ N is such that 1 ∈ S and for each n ∈ S 0 we also have n ∈ S, then S = N.) Let S ⊆ N be so that 1 ∈ S and so that for every n ∈ S we have that n0 ∈ S.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers But by deﬁnition of S, we also have S ⊆ N. Hence S = N.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 4. (If S ⊆ N is such that 1 ∈ S and for each n ∈ S 0 we also have n ∈ S, then S = N.) Let S ⊆ N be so that 1 ∈ S and so that for every n ∈ S we have 0 that n ∈ S. Because S is a successor set, by deﬁnition of N we conclude N ⊆ S.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Hence S = N.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 0 0 (For all m,n ∈ N if m = n , then m = n.) We ﬁrst use part 4 to prove that every element of n ∈ N is a subset of n. Let S := n ∈ N : [∀m ∈ n : m ⊆ n] . Trivially, {/0} ∈ S, that is, 1 ∈ S. For n ∈ S we have n0 = n ∪ {n}. Hence, if m ∈ n0, then m = n ⊆ n0 or m ∈ n, which means m ⊆ n ⊆ n0.

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Proof of part 5.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers We ﬁrst use part 4 to prove that every element of n ∈ N is a subset of n. Let S := n ∈ N : [∀m ∈ n : m ⊆ n] . Trivially, {/0} ∈ S, that is, 1 ∈ S. For n ∈ S we have n0 = n ∪ {n}. Hence, if m ∈ n0, then m = n ⊆ n0 or m ∈ n, which means m ⊆ n ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Proof of part 5. (For all m,n ∈ N if m = n , then m = n.)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Let S := n ∈ N : [∀m ∈ n : m ⊆ n] . Trivially, {/0} ∈ S, that is, 1 ∈ S. For n ∈ S we have n0 = n ∪ {n}. Hence, if m ∈ n0, then m = n ⊆ n0 or m ∈ n, which means m ⊆ n ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Proof of part 5. (For all m,n ∈ N if m = n , then m = n.) We ﬁrst use part 4 to prove that every element of n ∈ N is a subset of n.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Trivially, {/0} ∈ S, that is, 1 ∈ S. For n ∈ S we have n0 = n ∪ {n}. Hence, if m ∈ n0, then m = n ⊆ n0 or m ∈ n, which means m ⊆ n ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Proof of part 5. (For all m,n ∈ N if m = n , then m = n.) We ﬁrst use part 4 to prove that every element of n ∈ N is a subset of n. Let S := n ∈ N : [∀m ∈ n : m ⊆ n] .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers For n ∈ S we have n0 = n ∪ {n}. Hence, if m ∈ n0, then m = n ⊆ n0 or m ∈ n, which means m ⊆ n ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Proof of part 5. (For all m,n ∈ N if m = n , then m = n.) We ﬁrst use part 4 to prove that every element of n ∈ N is a subset of n. Let S := n ∈ N : [∀m ∈ n : m ⊆ n] . Trivially, {/0} ∈ S, that is, 1 ∈ S.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Hence, if m ∈ n0, then m = n ⊆ n0 or m ∈ n, which means m ⊆ n ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Proof of part 5. (For all m,n ∈ N if m = n , then m = n.) We ﬁrst use part 4 to prove that every element of n ∈ N is a subset of n. Let S := n ∈ N : [∀m ∈ n : m ⊆ n] . Trivially, {/0} ∈ S, that is, 1 ∈ S. For n ∈ S we have n0 = n ∪ {n}.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers or m ∈ n, which means m ⊆ n ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers , which means m ⊆ n ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers ⊆ n0.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m}

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers , which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers By the above, n ⊆ m and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers and m ⊆ n, that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers , that is, m = n, contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

0 0 Now let m,n ∈ N with m = n be arbitrary but ﬁxed. Then m ∪ {m} = m0 = n0 = n ∪ {n}. Suppose for a contradiction that m 6= n. Then {n} 6= {m}, which implies n ∈ m and m ∈ n. By the above, n ⊆ m and m ⊆ n

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers , contradiction.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 1. The Peano Axioms are derived from the axioms of set theory. 2. But axioms usually are given, not proved. 3. The Peano Axioms are a nice intermediate stage in our construction of the number systems from set theory. 4. Using them as the basis for further study allows us to worry less about sets. 5. Historically, the Peano Axioms were found before Russell’s Paradox and before the Zermelo-Fraenkel axioms for set theory.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 2. But axioms usually are given, not proved. 3. The Peano Axioms are a nice intermediate stage in our construction of the number systems from set theory. 4. Using them as the basis for further study allows us to worry less about sets. 5. Historically, the Peano Axioms were found before Russell’s Paradox and before the Zermelo-Fraenkel axioms for set theory.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 1. The Peano Axioms are derived from the axioms of set theory.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers , not proved. 3. The Peano Axioms are a nice intermediate stage in our construction of the number systems from set theory. 4. Using them as the basis for further study allows us to worry less about sets. 5. Historically, the Peano Axioms were found before Russell’s Paradox and before the Zermelo-Fraenkel axioms for set theory.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 1. The Peano Axioms are derived from the axioms of set theory. 2. But axioms usually are given

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 3. The Peano Axioms are a nice intermediate stage in our construction of the number systems from set theory. 4. Using them as the basis for further study allows us to worry less about sets. 5. Historically, the Peano Axioms were found before Russell’s Paradox and before the Zermelo-Fraenkel axioms for set theory.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 1. The Peano Axioms are derived from the axioms of set theory. 2. But axioms usually are given, not proved.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 4. Using them as the basis for further study allows us to worry less about sets. 5. Historically, the Peano Axioms were found before Russell’s Paradox and before the Zermelo-Fraenkel axioms for set theory.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 5. Historically, the Peano Axioms were found before Russell’s Paradox and before the Zermelo-Fraenkel axioms for set theory.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 1. The Peano Axioms are derived from the axioms of set theory. 2. But axioms usually are given, not proved. 3. The Peano Axioms are a nice intermediate stage in our construction of the number systems from set theory. 4. Using them as the basis for further study allows us to worry less about 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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 6. Compare with driving a car: A driver has to deal with a different axiomatic system (driving instructions) than a mechanic (engine function). 7. We cannot (and should not) think of all the engine functions as we drive (too distracting). 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers A driver has to deal with a different axiomatic system (driving instructions) than a mechanic (engine function). 7. We cannot (and should not) think of all the engine functions as we drive (too distracting). 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 6. Compare with driving a car:

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers (driving instructions) than a mechanic (engine function). 7. We cannot (and should not) think of all the engine functions as we drive (too distracting). 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 6. Compare with driving a car: A driver has to deal with a different axiomatic system

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers than a mechanic (engine function). 7. We cannot (and should not) think of all the engine functions as we drive (too distracting). 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 6. Compare with driving a car: A driver has to deal with a different axiomatic system (driving instructions)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers (engine function). 7. We cannot (and should not) think of all the engine functions as we drive (too distracting). 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 6. Compare with driving a car: A driver has to deal with a different axiomatic system (driving instructions) than a mechanic

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 7. We cannot (and should not) think of all the engine functions as we drive (too distracting). 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 6. Compare with driving a car: A driver has to deal with a different axiomatic system (driving instructions) than a mechanic (engine function).

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers , and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key. Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers Knowing that we need the engine to turn over is helpful when starting a car with electrical problems.

Axiom of Inﬁnity Natural Numbers Axiomatic Systems

Different Levels of Axiomatic Systems 6. Compare with driving a car: A driver has to deal with a different axiomatic system (driving instructions) than a mechanic (engine function). 7. We cannot (and should not) think of all the engine functions as we drive (too distracting). 8. The two still connect: Engine function allows us to drive, and some knowledge about the function of the engine can be helpful. 9. For example, to start, the engine must turn over. The handcrank from the really old movies has been replaced with an electric motor that cranks as we turn the key.

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

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Axiom of Inﬁnity and The Natural Numbers