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> a well-founded relation ⊏ is a well-ordering if and only if it relates every distinct pair of elements, i.e. x1≠x2 implies either x1⊏x2 or x2⊏x1. The author
by kccqzy 2y ago
> a well-founded relation ⊏ is a well-ordering if and only if it relates every distinct pair of elements, i.e. x1≠x2 implies either x1⊏x2 or x2⊏x1.
The author here explains well-founded relation prior to this. I want to add an example here. Consider the set of positive real numbers. If we use the natural ordering < then we have a total order. But that's not a well-ordering. If x were to be the least element you could always divide it by 2 to get a lesser element. So there's no least element in the set of positive reals using the conventional < ordering. However, the amazing thing is that assuming axiom of choice, such a well ordering exists.
- GrantMoyer 2y agoI think maybe you're overlooking "a well-founded relation ⊏ is a well-ordering …". < on the reals is not well-founded. Remember, a well-founded relation requires all leftward chains to be finite. Together with the rest of the given definition, it implies a unique least element.
- kccqzy 2y agoYes you are right. My bad. I edited my original comment.
- lupire 2y agoRecall that the set R of real numbers R is defined as the set of numbers that can be shown to exist, plus an uncountable collection of "numbers" that are assumed to be near the set of numbers that can be shown to exist. It's not amazing that assuming something unnatural (Uncountable Choice) gives an equally unnatural consequence. The set R of "Real" numbers is a fantastical "object" that has fantastical properties. It can do apparently impossible things because ZFC flat out assumes that it can do apparently impossible things.
- immibis 2y agoIIRC the real numbers are, roughly speaking, more-or-less isomorphic to infinite sequences of digits. If you believe real numbers are fantastical, I suppose you also believe infinite sequences are fantastical? We can't have one in the universe, because we'd run out of atoms, but we can't have circles either and that is rarely complained about.
- lupire 2y agoAn infinite computable sequence is fine. Making "choices" from an uncountable collection of uncomputable sequences is not fine. I'm embarrassed to have to write that out.
- kccqzy 2y agoHistorically the definition of computability came quite a bit later than ZFC. Zermelo might have formulated his set theory differently if computability came first.
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- martin293 2y agoWell how is it constructed?
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- kccqzy 2y agoThe existence proof uses the axiom of choice so it's not a constructive proof. You cannot deduce how it's constructed from its existence.