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

[@lucianozaffaina9853]

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

[@LetsLearnNemo]

Yes, this is one of the beautiful properties about the set of real numbers, in the sense that if a subset of R is bounded above, a supremum always exists. For maximal element existence, this is a bit harder to target for sets that are dense in R. Chains are typically (from what I've seen) only used in countable subsets of sets (although one could extend to uncountable); but these nuances are not of vital importance for what we do with them in the series, just the sup property for sets bounded above is a major piece of machinery for the realm of real analysis. Posets and chains is more useful for building structure on sets that may not have a total ordering (toset) property like R does. Just my surface thoughts, several more things I could add here. Feel free to reach out for specifics :)

[@juanaquino7432]

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.

[@LetsLearnNemo]

Thank you for your kind words! And yes, you are correct. Retrospectively, I would prefer to say that a relation R is reflexive, provided that for any x chosen from S, (x,x) must belong to R; perhaps this is a safer way.