(RA02) Axiom of Choice, Zorn's Lemma, and the Well Ordering Principle

In this video, we explore a few options for our foundational axiom(s). Note that a set of axioms usually is chosen to be the smallest number of assumptions that one needs in order to construct as much as you can from. In particular, we discuss the Axiom of Choice, Zorn's Lemma, and the Well Ordering Principle, all of which are equivalent to one another (i.e. any one of them implies another). During their explorations, we discuss partially ordered sets, totally ordered sets, choice functions, and the differences/similarities between upper bounds, least upper bounds, maximal elements, and maximum elements.

*--The Let's Learn, Nemo Community--*

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[lucianozaffaina]

What if y_w does not belong to S? Example S=[0, 1). Upper bound 1 does not belong to S. In this case every chain has an upper bound but S has non Maximal element

[juanaquino]

You are an excelent teacher. Thank you very much in your clear presentation of ideas. I have a small doubt. In 8:10 you stated the reflexive property somewhat different of the definition. A relation R on a set S, R⊆S×S is reflexive if and only if for all x∈S we have that xRx. The if and only if statement in your definition is always true since it adresses the definition of aRb notation. In the sense that (aRb) is a shorthand of (a, b)∈R. So xRx if and only if (x, x)∈R si indeed always true but not the definition of the reflexive property.