Certainly the most familiar wellordered set is 0,1,2, the set of nonnegative integers. So the well ordering principle is the following statement. In mathematics, the well ordering principle states that every nonempty set of positive integers contains a least element. When proving that a wellordered set satisfies the strong induction principle, the ordering of the set is supposed to be given, and to be a strict total order. The wellordering principle of the natural numbers mathonline.
However, because of the very way in which we constructed the set of natural numbers and its arithmetic, we deduced, in class. The set ais a subset of n, and we proved that subsets of well ordered sets are still well ordered in a homework problem. Intuitively, the notion of order among elements of a set, x, captures the fact some elements are bigger than others, perhaps more important, or. Every nonempty set of positive integers contains a smallest member. Strong induction and wellordering niloufar shafiei. So in this section we shall embark upon a systematic study of well ordered sets. The wellordering theorem can be used to solve all sorts of mathematical problems but it is a powerful medicine.
Cs 6110 lecture 7 wellfounded induction 6 february 20. If in theorem 3 we take xto be a closed subinterval of r, we recover real induction and its variants as discussed above. Every set of natural numbers except the empty set has a smallest element. A set of real numbers s is well ordered if every nonempty subset of s has a least element. Any set of integers larger than a given number is well ordered. In addition to the insufficiency of our axioms with respect to particular extensive sets, the general method of definition by transfinite induction and, in particular, the proof that to every wellordered set there is a corresponding ordinal number 5. A set of real numbers s is wellordered if every nonempty subset of s has a least element. A well ordering is a consistent choice function so while my last comment was meant to have a certain punch to it, i should probably say a little more. So, the well ordering principle captures something special about the nonnegative integers. In this worksheet we will show that they are equivalent, so you only need assume one of the two axioms. We begin our look through abstract algebra with a rather simple theorem regarding the set of natural numbers known as the wellordering principle of the natural numbers. The set ais a subset of n, and we proved that subsets of wellordered sets are still wellordered in a homework problem. The wellordering principle also implies the principle of mathematical induction via the following theorem rules. There is some m 2 t because t is nonempty, so that t contains a natural number m, which is just m itself.
The wellordering property states that every nonempty set of nonnegative integers has a least element. A nonempty subset s of r is wellordered if every nonempty subset of s has a smallest element. If we assume the axiom of choice which we will in this talk, we can prove the wellordering theorem. However, the set of integers with our usual ordering on it is not wellordered, neither is the set of rational numbers, nor the set of all positive rational numbers. Pdf reverse mathematics and wellordering principles. Applying this to halfclosed intervals as above gives real induction.
Today we discuss indexed sets and the well ordering principle. If j is a well ordered set, no section of j has the order type of j. Assume strong induction is not valid proof by contradiction, so n pn. Induction and the wellordering principle the principle of mathematical induction and the wellordering principle are considered axioms for the construction of the usual number systems meaning that we must assume them to be true. This is called, by mathematicians of various stripes, wellfounded induction or noetherian induction. Jan 22, 2020 any set of integers larger than a given number is well ordered. More generally, a property concerning the positive integers that is true for \n1\, and that is true for all integers up to. We can, for example, generalize the method of induction so that it can be used in proofs about sets of objects more general than a set of integers greater than some integer a. Induction and the well ordering principle the principle of mathematical induction and the well ordering principle are considered axioms for the construction of the usual number systems meaning that we must assume them to be true. We discuss examples that are not well ordered and not totally ordered. Consider the following set which we define to be the set of natural numbers.
The well ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. Even though these examples seem silly, both trivial and vacuous proofs are often used. If we assume the axiom of choice which we will in this talk, we can prove the well ordering theorem. Wellordering principle for natural numbers states that every subset of n other than. This is an axiom, meaning we will accept it without demanding a proof. Every nonempty subset s s s of the positive integers has a least element note that this property is not true for subsets of the integers in which. As a result, there are some integers that are not contained in \s. How to prove the well ordering principle using induction.
So now suppose that c x1 set of maximal elements of p. Recall that a simply ordered set x is said to be well ordered if every nonempty subset y has a smallest element, i. Jan 29, 2012 wellordering principle for natural numbers states that every subset of n other than. The well ordering principle of the natural numbers. A principle of mathematical induction for partially. In mathematics, a well order or well ordering or well order relation on a set s is a total order on s with the property that every nonempty subset of s has a least element in this ordering. The principle of mathematical induction is equivalent to the priciniple of strong induction and both are equivalent to the well ordering principle.
The well ordering property states that every nonempty set of nonnegative integers has a least element. Chapter 5 partial orders, lattices, well founded orderings. The symbol \\mathbbz\ denotes the set of all integers. As we will see, the ability to use induction is intimately related to a very special property of partial orders known as wellfoundedness. Also, applying it to a well ordered set recovers transfinite induction exactly as it is usually stated i. The wellordering theorem anyset x canbewellordered. The well ordering principle and mathematical induction. It can also be stated for all sets, not just sets of integers and is. Chapter viii ordered sets, ordinals and transfinite methods. The fact that this set is wellordered is exactly what makes traditional induction work. In other words, the set of positive integers is well ordered by its natural or magnitude order in which. Principle of transfinite induction we say that a relation r well orders a set x if r linearly orders x and every nonempty subset of x contains a first element with respect to the relation r. It can also be stated for all sets, not just sets of integers and is related to zorns lemma and the axiom of choice.
The validity of both the principle of mathematical induction and strong induction follows from a fundamental axiom of the set of integers, the well ordering property. A wellorder is a particular kind of p p coalgebra, but the punch line was that the category of wellorders and initial segment inclusions is a full subcategory of the category of p p. The discussion of cardinalities in chapter 2 suggests that the most interesting thing about them is their order relation, namely that any set of cardinalities forms a well ordered set. Since r is the unique dedekind complete ordered eld. This proposition claims that if a set has a lower bound which is a real number, then this set is well ordered. As far as i can see it should be called artinian induction. In general, a set such as n with some order well ordered if any nonempty subset has a least element. Further, this proposition implies the principle of mathematical induction. In other words, the set of positive integers is wellordered by its natural or magnitude order in which.
This is why we often prove one from the other, and vice versa. Why do you suppose mathematicians are willing to accept this. However, the set of integers with our usual ordering on it is not well ordered, neither is the set of rational numbers, nor the set of all positive rational numbers. A set s is well ordered if every subset has a least element. Proofs the well ordering principle duke computer science. Nov 29, 2014 today we discuss indexed sets and the well ordering principle. The most familiar example of a wellordered set is and it is the wellordering property th at lets us do mathematical induction in in this chapter we will see longer well ordered sets and these will give us a new proof method called transfinite induction. The principle of wellfounded induction on the relation. The set s together with the well order relation is then called a well ordered set. In your textbook, there is a proof for how the wellordering principle implies the validity of mathematical induction. At least if we assume the natural numbers are a structure which satisfies some basic axioms. While the well ordering principle may seem obvious, its hard to see offhand why it is useful.
Assume also that \s\ is not the set of all positive integers. Even though these examples seem silly, both trivial and vacuous proofs are often used in mathematical induction, a widelyused. If a set is well ordered or even if it merely admits a well founded relation, the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set. An ordered set is said to be wellordered if each and every nonempty subset has a smallest or least element. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or. Wellorderings are useful because they allow proofs by induction. The most familiar example of a well ordered set is and it is the well ordering property th at lets us do mathematical induction in in this chapter we will see longer well ordered sets and these will give us a new proof method called transfinite induction. We begin our look through abstract algebra with a rather simple theorem regarding the set of natural numbers known as the well ordering principle of the natural numbers. No property of strict total orders needs to be proved. In mathematics, the wellordering principle states that every nonempty set of positive integers contains a least element. This is also a set of natural numbers, to which we will apply the wellordered axiom. Well ordering and induction mathematics libretexts. A well ordering is a consistent choice function the n. Proof of strong induction using wellordering principle.
The following important fact is called the well ordering principle. When proving that a well ordered set satisfies the strong induction principle, the ordering of the set is supposed to be given, and to be a strict total order. We will show how we can directly use the well ordering property in proofs. The validity of both the principle of mathematical induction and strong induction follows from a fundamental axiom of the set of integers, the wellordering property. Strong induction and well ordering york university. The theory of ordinals is closely related to the theory of well ordered sets see section 10 of munkres.
Ordinal numbers and the wellordering theorem cornell. Well ordering theorem every set can be well ordered. Every nonempty set of nonnegative integers has a least element. Very often instead of writing, say, x, y belongs to r we write xry. There are some sets we can easily see are well ordered. There are some sets we can easily see are wellordered. The wellordering principle says that the positive integers are wellordered. The well ordering principle says that the positive integers are well ordered. Feb 29, 2020 we use the well ordering principle to prove the first principle of mathematical induction. We will show how we can directly use the wellordering property in proofs.
Show that strong induction is a valid method of proof by showing that it follows from the wellordering property. In other words, either sc has a smallest element or else it is the empty set. An ordered set is said to be well ordered if each and every nonempty subset has a smallest or least element. A principle of mathematical induction for partially ordered. Well ordering and lexicographical orde r we are often interested in sets of objects that are equipped with an order relation that satisfies certain properties. J e thlat is order preserving and whose image is e or a section of e. Since r is the unique dedekind complete ordered eld, we deduce. Recall that the set of natural numbers with the order well ordered. Taking xto be any well ordered set, we recover trans nite induction and, in particular, \ordinary mathematical induction. Jan 29, 2015 we discuss examples that are not well ordered and not totally ordered. In this chapter we will see longer well ordered sets and these will give us a new proof method called. The wellordering theorem can be used to solve all sorts of mathematical problems but it is a. For example, the set n of natural numbers is well ordered by its natural ordering. So the wellordering principle is the following statement.
There is a wellknown principle of induction on partially ordered sets satisfying the descending chain condition, or equivalently, in which every nonempty subset has a minimum. The principle of mathematical induction is equivalent to the priciniple of strong induction and both are equivalent to the wellordering principle. Taking xto be any wellordered set, we recover trans nite induction and, in particular, \ordinary mathematical induction. Consider the complementary set scwhose elements are the natural numbers that are not elements of s.
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