Concrete non-measurable sets

[Kreizhn]

"Concrete" non-measurable sets

I've had Vitali's proof of the existence of non-(Lebesgue) measurable sets branded into the side of my brain over the years. However, the proof always critically relies on evoking the axiom of choice. Has anybody every demonstrated a non-AoC construction of a non-measurable set? Or do the intuitionist logicians just avoid measure theory all together?

 

[micromass]

Hi Kreizhn!

The axiom of choice is necessary for the existence of non-measurable sets. If you don't adopt the axiom of choice (or a similar principle), then it can happen that all sets are measurable! There is a current investigation of some theories in which all sets are measurable, so it certainly can happen!

 

[Kreizhn]

Good to know. Then does using AoC necessarily preclude the ability to construct non-measurable sets that are anything but "existential?"

For example, (unless it's just been so long since I've done set theory) I imagine there are times when we must use the AoC in general, but for special cases we could explicitly construct a choice function. Are there special cases wherein this can be done for non-measurable sets? Maybe that's just a silly question.

 

[micromass]

Good to know. Then does using AoC necessarily preclude the ability to construct non-measurable sets that are anything but "existential?"

For example, (unless it's just been so long since I've done set theory) I imagine there are times when we must use the AoC in general, but for special cases we could explicitly construct a choice function. Are there special cases wherein this can be done for non-measurable sets? Maybe that's just a silly question.

I don't quite understand what you're asking. Do you want to construct a choice function for a non-measurable set?? This is not possible, everything you do with non-measurable sets involves the axiom of choice. It is not possible to construct a choice function for them.

 

[Kreizhn]

I guess what I'm thinking is that AoC always guarantees the existence of a choice function, without explicitly defining it. However, there are instances in which AoC is not required to give the choice function.

But then I guess the answer is that if we must always use AoC, then it is impossible to explicitly construct a choice function, since then AoC would not be necessary to construct nonmeasurable sets, and you have just told me that it is.