By mounting a philosophical historical thought experiment, I argue that our attitude toward the continuum hypothesis could easily have been very different than it is.
“The answers provided by PD are better and more uniform and so forth than the corresponding answers provided by V=L, which leads to a vision in contrast of set theory as the land of counterexamples and bad news, while in PD so many things work smoothly.”
This aesthetic judgment may be good, but I don’t know because I have no idea what you mean by “smoothly” vs “bad news”.
As a third alternative, if you assume a Real Valued Measurable Cardinal, do the answers to these questions in descriptive set theory come out more like PD or more like V=L, and how do they compare according to the aesthetic criteria you are using?
V=L is viewed as bad news since it provides a Delta^1_2 definable well-ordering of the reals, and with this one can construct low-level definable counterexamples exhibiting all the classic troubles with the axiom of choice. Low-level definable non-measurable sets, definable Banach-Tarski sets, and so forth. Whereas with PD one has that every definable set of reals (projectively definable) is Lebesgue measurable and has the property of Baire and so forth, and this is what I meant by the "smooth" theory. Under PD every projectively definable set of reals obeys the continuum hypothesis.
If the continuum is a real-valued measurable cardinal, then this implies V ≠ L of course, and perhaps the axiom is a little closer to PD, since AD after all implies that omega_1 is a measurable cardinal and also omega_2 and more. PD does not imply that, but there is the residual of this for definable sets of reals.
I wasn't taking sides, so much as expressing the standard view. But meanwhile, I do view V=L as a means of producing counterexamples, without a nice unifying theory, at the level of the reals. Of course, V=L does provide a unifying picture (if limiting) of the nature of set existence in the transfinite hierarchy that defines it.
Thanks for this. Not a pluralism qn: Given Prof. Woodins more recent phil orientated lectures, (i.e justifying Additional axioms https://youtu.be/WaSBt0RZBRY?si=VUHT2IRGcRT8R0tj) and the comments on the usefulness of ZFC+PD on closing second order PA, how does that fit with the Categorization you talk of? I think that a world of ZFC+PD+CH is possible (i.e. PD is compatible with CH) but ZF+AD is not … is the process to find the point (in V) at which we satisfy the ability for CH to remain compatible (or independent, and thus able to be assumed as an axiom) but also retain the benefits something like PD provides (in its own usefullness wrt PA)… akin to woodin’s own Ultimate L ?
Yes, ZFC+PD+CH is equiconsistent with ZFC+PD, since you can force CH without adding reals, and this will therefore preserve PD, since it doesn't change the projective sets or the strategies. I guess the arguments one hears for PD are not generally based on categoricity, however, but on leading us to the right answers to the questions that we have. The answers provided by PD are better and more uniform and so forth than the corresponding answers provided by V=L, which leads to a vision in contrast of set theory as the land of counterexamples and bad news, while in PD so many things work smoothly.
“The answers provided by PD are better and more uniform and so forth than the corresponding answers provided by V=L, which leads to a vision in contrast of set theory as the land of counterexamples and bad news, while in PD so many things work smoothly.”
This aesthetic judgment may be good, but I don’t know because I have no idea what you mean by “smoothly” vs “bad news”.
As a third alternative, if you assume a Real Valued Measurable Cardinal, do the answers to these questions in descriptive set theory come out more like PD or more like V=L, and how do they compare according to the aesthetic criteria you are using?
V=L is viewed as bad news since it provides a Delta^1_2 definable well-ordering of the reals, and with this one can construct low-level definable counterexamples exhibiting all the classic troubles with the axiom of choice. Low-level definable non-measurable sets, definable Banach-Tarski sets, and so forth. Whereas with PD one has that every definable set of reals (projectively definable) is Lebesgue measurable and has the property of Baire and so forth, and this is what I meant by the "smooth" theory. Under PD every projectively definable set of reals obeys the continuum hypothesis.
If the continuum is a real-valued measurable cardinal, then this implies V ≠ L of course, and perhaps the axiom is a little closer to PD, since AD after all implies that omega_1 is a measurable cardinal and also omega_2 and more. PD does not imply that, but there is the residual of this for definable sets of reals.
Why is it a bad thing rather than a good thing that counterexamples that definitely exist are actually definable?
I wasn't taking sides, so much as expressing the standard view. But meanwhile, I do view V=L as a means of producing counterexamples, without a nice unifying theory, at the level of the reals. Of course, V=L does provide a unifying picture (if limiting) of the nature of set existence in the transfinite hierarchy that defines it.
Thanks for this. Not a pluralism qn: Given Prof. Woodins more recent phil orientated lectures, (i.e justifying Additional axioms https://youtu.be/WaSBt0RZBRY?si=VUHT2IRGcRT8R0tj) and the comments on the usefulness of ZFC+PD on closing second order PA, how does that fit with the Categorization you talk of? I think that a world of ZFC+PD+CH is possible (i.e. PD is compatible with CH) but ZF+AD is not … is the process to find the point (in V) at which we satisfy the ability for CH to remain compatible (or independent, and thus able to be assumed as an axiom) but also retain the benefits something like PD provides (in its own usefullness wrt PA)… akin to woodin’s own Ultimate L ?
Yes, ZFC+PD+CH is equiconsistent with ZFC+PD, since you can force CH without adding reals, and this will therefore preserve PD, since it doesn't change the projective sets or the strategies. I guess the arguments one hears for PD are not generally based on categoricity, however, but on leading us to the right answers to the questions that we have. The answers provided by PD are better and more uniform and so forth than the corresponding answers provided by V=L, which leads to a vision in contrast of set theory as the land of counterexamples and bad news, while in PD so many things work smoothly.