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.

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.

While some of the parent turns out not to hold, it helped me to find out what the theory really says (now that I have time).

2vi21maobk9vp14y

Separate concern: Why constructible real numbers are only finitely higher than Q? Cannot it be that there are some elements of (say) 2^Q that cannot be pinpointed until a much higher ordinal? Of course, there is still a formula that specifies a high enough ordinal to contain all members of R that are actually constructible.

0vi21maobk9vp14y

Let's see. Assume measurability axiom - every subset of R has Lebesgue measure. As we can use the usual construction of unmeasurable set on L intersect R, our only escape option is that it has zero measure. So if we assume measurability, L intersect R is a dense zero-measure subset, just like Q. These are the reals we can know individually, but not the reals-as-a-whole that we know...