Stupid Questions Open Thread Round 2

OpenThreadGuy

From Costanza's original thread (entire text):

This is for anyone in the LessWrong community who has made at least some effort to read the sequences and follow along, but is still confused on some point, and is perhaps feeling a bit embarrassed. Here, newbies and not-so-newbies are free to ask very basic but still relevant questions with the understanding that the answers are probably somewhere in the sequences. Similarly, LessWrong tends to presume a rather high threshold for understanding science and technology. Relevant questions in those areas are welcome as well. Anyone who chooses to respond should respectfully guide the questioner to a helpful resource, and questioners should be appropriately grateful. Good faith should be presumed on both sides, unless and until it is shown to be absent. If a questioner is not sure whether a question is relevant, ask it, and also ask if it's relevant.

Meta:

How often should these be made? I think one every three months is the correct frequency.
Costanza made the original thread, but I am OpenThreadGuy. I am therefore not only entitled but required to post this in his stead. But I got his permission anyway.

From Costanza's original thread (entire text):

This is for anyone in the LessWrong community who has made at least some effort to read the sequences and follow along, but is still confused on some point, and is perhaps feeling a bit embarrassed. Here, newbies and not-so-newbies are free to ask very basic but still relevant questions with the understanding that the answers are probably somewhere in the sequences. Similarly, LessWrong tends to presume a rather high threshold for understanding science and technology. Relevant questions in those areas are welcome as well. Anyone who chooses to respond should respectfully guide the questioner to a helpful resource, and questioners should be appropriately grateful. Good faith should be presumed on both sides, unless and until it is shown to be absent. If a questioner is not sure whether a question is relevant, ask it, and also ask if it's relevant.

Meta:

How often should these be made? I think one every three months is the correct frequency.
Costanza made the original thread, but I am OpenThreadGuy. I am therefore not only entitled but required to post this in his stead. But I got his permission anyway.

Does ZF assert the existence of an actual formula, that one could express in ZF with a finite string of symbols, defining a well-ordering on the-real-numbers-as-we-know-them? I know it 'proves' the existence of a well-ordering on the set we'd call the real numbers if we endorsed the statement "V=L". I want to know about the nature of that set, and how much ZF can prove without V=L or any other form of choice.

Yes. In ZF one can construct an explicit well-ordering of L(alpha) for any alpha; see e.g. Kunen ch VI section 4. The natural numbers are in L(omega) and so the constructible real numbers are in L(omega+k) for some finite k whose value depends on exactly how you define the real numbers; so a well-ordering of L(omega+k) gives you a well ordering of R intersect L.

I'm not convinced that R intersect L deserves the name of "the-real-numbers-as-we-know-them", though.

1Incorrect14y

Is this a correct statement of what a well-ordering of R is? %20\land%20\forall%20B%20(B%20\subseteq%20\mathbb{R}%20\land%20B%20\neq%20\emptyset%20\implies%20\exists%20x%20\in%20B%20\forall%20y%20\in%20B%20((x%20\preceq%20y)%20\in%20A)))%0A%0A\forall%20A%20\forall%20B%20(\text{total-order}(A,%20B)%20\iff%20\forall%20a%20\forall%20b%20\forall%20c(a,%20b,%20c%20\in%20B%20\implies%20(((a%20\preceq%20b)%20\in%20A%20\lor%20(b%20\preceq%20a)%20\in%20A)%20\land%20((a%20\preceq%20b)%20\in%20A%20\land%20(b%20\preceq%20a)%20\in%20A%20\implies%20a%20=%20b)%20\land%20((a%20\preceq%20b)%20\in%20A%20\land%20(b%20\preceq%20c)%20\in%20A%20\implies%20(a%20\preceq%20c)%20\in%20A)))))

8Thomas14y

ZF does not imply well ordering of R, ZFC does. ZF with the Axiom of Choice. Strictly speaking.

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