Misha comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong

8 Post author: Unnamed 25 August 2011 02:17AM

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Comment author: [deleted] 12 September 2011 03:06:25PM 0 points [-]

That's intriguing. Do you have any examples of what people actually work with?

Comment author: TobyBartels 12 September 2011 06:55:11PM 4 points [-]

Sniffnoy may have more examples, but here are some that I know:

  • Every subset of the real line is Lebesgue-measurable.
  • Every subset of the real line has the Baire property (in much the same vein as the preceding one).
  • The axiom of determinacy (a statement in infinitary game theory).

Adding the first two to ZF + DC (dependent choice) is consistent (assuming that ZFC + Con(ZFC) is consistent, as just about everybody believes), and this gives a "dream universe" for analysis in which, for example, any everywhere-defined linear operator between Hilbert spaces is bounded.

Comment author: Eugine_Nier 13 September 2011 08:02:22AM 4 points [-]

Adding the first two to ZF + DC (dependent choice) is consistent (assuming that ZFC + Con(ZFC) is consistent, as just about everybody believes)

This isn't quite right. The consistency of ZF + DC + "every subset of R is Lebesgue measurable" is equivalent to the consistency of an inaccessible cardinal, which is a much stronger assumption then the consistency of ZFC + Con(ZFC).

Comment author: TobyBartels 17 September 2011 08:49:50PM 2 points [-]

Sorry, my mistake. Still, set theorists usually believe this.

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