Well-order

Order theory | Wellfoundedness | Ordinal numbers | Binary relations

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 non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see below for an example). A well-ordered set S contains for every subset T with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of T in S. If ≤ is a non-strict well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a well-founded strict total order. The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible. Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. 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. The observation that the natural numbers are well ordered by the usual less-than relation is commonly called the well-ordering principle (for natural numbers). (Wikipedia).

Set Theory 1.4 : Well Orders, Order Isomorphisms, and Ordinals

In this video, I introduce well ordered sets and order isomorphisms, as well as segments. I use these new ideas to prove that all well ordered sets are order isomorphic to some ordinal. Email : fematikaqna@gmail.com Discord: https://discord.gg/ePatnjV Subreddit : https://www.reddit.com/r/

From playlist Set Theory

Well-Ordering and Induction: Part 1

This was recorded as supplemental material for Math 115AH at UCLA in the spring quarter of 2020. In this video, I prove the equivalence of the principle of mathematical induction and the well-ordering principle.

From playlist Well Ordering and Induction

Laws of Arithmetic (3 of 3: The Associative Law)

More resources available at www.misterwootube.com

From playlist Fractions, Decimals and Percentages

Orders on Sets: Part 1 - Partial Orders

This was recorded as supplemental material for Math 115AH at UCLA in the spring quarter of 2020. In this video, I discuss the concept and definition of a partial order.

From playlist Orders on Sets

Definition of the Order of an Element in a Group and Multiple Examples

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of the Order of an Element in a Group and Multiple Examples

From playlist Abstract Algebra

Working with Functions (1 of 2: Notation & Terminology)

More resources available at www.misterwootube.com

From playlist Working with Functions

Zorn's Lemma, The Well-Ordering Theorem, and Undefinability (Version 2.0)

Zorn's Lemma and The Well-ordering Theorem are seemingly straightforward statements, but they give incredibly mind-bending results. Orderings, Hasse Diagrams, and the Ordinals / set theory will come up in this video as tools to get a better view of where the "proof" of Zorn's lemma comes f

From playlist The New CHALKboard

Discrete Math - 5.2.1 The Well-Ordering Principle and Strong Induction

In this video we introduce the well-ordering principle and look and one proof by strong induction. Textbook: Rosen, Discrete Mathematics and Its Applications, 7e Playlist: https://www.youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNUNoej-vY5Rz

From playlist Discrete Math I (Entire Course)

Set Theory (Part 11): Ordering of the Natural Numbers

Please feel free to leave comments/questions on the video and practice problems below! In this video, we utilize the definition of natural number to speak of ordering on the set of all natural numbers. In addition, the well-ordering principle and trichotomy law are proved.

From playlist Set Theory by Mathoma

How To Create Supply Chain dApp For Order Accounting | Session 09 | #ethereum | #blockchain

Don’t forget to subscribe! In this project series, we will learn to create a supply chain dApp for order accounting. This series will cover all the details (resources, tools, languages, etc) that are necessary to build a complete and operational supply chain dApp over the Ethereum blockc

From playlist Create Supply Chain dApp For Order Accounting

How To Create Supply Chain dApp For Order Accounting | Session 04 | #ethereum | #blockchain

From playlist Create Supply Chain dApp For Order Accounting

Introduction to number theory lecture 23. Primitive roots.

This lecture is part of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 We show that every prime has a primitive root. The textbook is "An introduction to the the

From playlist Introduction to number theory (Berkeley Math 115)

Lagrange's Theorem -- Abstract Algebra 10

⭐Support the channel⭐ Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math My amazon shop: https://www.amazon.com/shop/michaelpenn 🟢 Discord: https://discord.gg/Ta6PTGtKBm ⭐my other channels⭐ Main Channel: https://www.youtube.

From playlist Abstract Algebra

Another Math of Infinity: Ordinals and "going past infinity"

As a follow up to the last video, here we introduce another math of infinity based on the thought experiment about infinite hotels from last time. Ordinals and well-orders in general give us a another way to think about infinite things; in terms of length and so in the context of ordinals

From playlist The CHALKboard 2022

How To Create Supply Chain dApp For Order Accounting | Session 03 | #ethereum | #blockchain

From playlist Create Supply Chain dApp For Order Accounting

How To Create Supply Chain dApp For Order Accounting | Session 05 | #ethereum | #blockchain

From playlist Create Supply Chain dApp For Order Accounting

Galois theory: Finite fields

This lecture is part of an online graduate course on Galois theory. We use the theory of splitting fields to classify finite fields: there is one of each prime power order (up to isomorphism). We give a few examples of small order, and point out that there seems to be no good choice for

From playlist Galois theory

Live CEOing Ep 266: Language Design in Wolfram Language

Watch Stephen Wolfram and teams of developers in a live, working, language design meeting. This episode is about Language Design in the Wolfram Language.

From playlist Behind the Scenes in Real-Life Software Design

Order of Operations

http://www.tabletclass.com explains the order of operations

From playlist Pre-Algebra

Well-order

Related pages