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.
Thank you for posting this, I really enjoyed the talk even as a relative layperson.
One question I had was how the lack of categoricity manifests when you don't have CH. Do the different sets which might qualify as the hyperreals differ in any properties we can pick out, or is it just that in such a model there fails to be an isomorphism between different sets which qualify as the hyperreals? If the latter, I'm surprised we care so much about getting to say "the" instead of "a".
In practice, we don't currently routinely assume CH, and yet people look into nonstandard analysis. They just work with some R*, and don't fuss about which one. Categoricity is not needed in the actual mathematical development, and this point was discussed in the Q & A with Marcus. What categoricity does provide is a philosophical justification of the subject, a unifying perspective on the nature of the subject--it helps us to see the subject as one thing, rather than as a bunch of different similar things that we cannot easily resolve. Imagine what doing real analysis would be like, if we knew that there were different mathematical structures and not just one version of the real numbers (up to isomorphism). It would impact how we think about real analysis in a deep way. And similarly, I argued, for the hyperreals when we don't have CH.
“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.
When I imagine what alien civilizations might assume in their math, I expect they would all have discovered set theory because there is something special and compelling about it and it is uniquely strong compared with other branches of math. I also expect that because of historical contingencies they could believe very incompatible things about third order arithmetic. They might assume something like V=L or something even more restrictive that implied no inaccessible or no standard model, they might assume PD or AD, they might assume RVM, etc. Discussions about axioms with them might have a religious character. But I can NOT imagine that they would disagree with us, or each other, about arithmetic! That is, there might be arithmetical statements about which there is disagreement whether those statements have been proven, but not arithmetical statements such that one civilization believes they are true and another believes they are false.
Another way of looking at this: Kronecker had it backwards if you are thinking about knowledge. Man can know the integers, all the rest can only be known by God.
Well, they might believe not only that there are no inaccessible cardinals, but they might believe that the assertion that they exist is actually inconsistent. This would be an arithmetic statement that they think is false, but most set theorists on Earth think is true. Or perhaps they think ZFC is inconsistent or even much weaker theories. And we know that there is in principle nothing dangerous about those beliefs, because we can prove that they are consistent relative to our theory, so we can't really prove them wrong. I think there is room for thinking that there could be genuine arithmetic disagreements.
I don’t think this is anywhere near as likely. No matter how strong their intuition was that there were no inaccessibles, after a century or so they would have come around to recognizing that this was something that needed its own axiom.
The ancients and medievals didn’t have the right attitude here, and found the counterintuitiveness of denying the parallel postulate so disturbing that they kept looking for refutations from the other Euclidean axioms for many centuries, but for a long time now people have been unperturbed when some axioms aren’t strong enough to imply another one.
Although I can’t think of any examples in the modern era of logic (1930’s onward) where it took many decades to discover an inconsistency in a system, there are examples of systems whose consistency could not be proven for a very long time, which is almost the same thing. The recent machine-verified proof of the consistency of Quine’s New Foundations system (announced by Randall Holmes a decade ago but only fully verified last year) is the best one I know of.
What’s the record for the amount of time between an axiom being suggested, and it being found to be inconsistent?
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.
Thank you for posting this, I really enjoyed the talk even as a relative layperson.
One question I had was how the lack of categoricity manifests when you don't have CH. Do the different sets which might qualify as the hyperreals differ in any properties we can pick out, or is it just that in such a model there fails to be an isomorphism between different sets which qualify as the hyperreals? If the latter, I'm surprised we care so much about getting to say "the" instead of "a".
In practice, we don't currently routinely assume CH, and yet people look into nonstandard analysis. They just work with some R*, and don't fuss about which one. Categoricity is not needed in the actual mathematical development, and this point was discussed in the Q & A with Marcus. What categoricity does provide is a philosophical justification of the subject, a unifying perspective on the nature of the subject--it helps us to see the subject as one thing, rather than as a bunch of different similar things that we cannot easily resolve. Imagine what doing real analysis would be like, if we knew that there were different mathematical structures and not just one version of the real numbers (up to isomorphism). It would impact how we think about real analysis in a deep way. And similarly, I argued, for the hyperreals when we don't have CH.
“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.
When I imagine what alien civilizations might assume in their math, I expect they would all have discovered set theory because there is something special and compelling about it and it is uniquely strong compared with other branches of math. I also expect that because of historical contingencies they could believe very incompatible things about third order arithmetic. They might assume something like V=L or something even more restrictive that implied no inaccessible or no standard model, they might assume PD or AD, they might assume RVM, etc. Discussions about axioms with them might have a religious character. But I can NOT imagine that they would disagree with us, or each other, about arithmetic! That is, there might be arithmetical statements about which there is disagreement whether those statements have been proven, but not arithmetical statements such that one civilization believes they are true and another believes they are false.
Another way of looking at this: Kronecker had it backwards if you are thinking about knowledge. Man can know the integers, all the rest can only be known by God.
Well, they might believe not only that there are no inaccessible cardinals, but they might believe that the assertion that they exist is actually inconsistent. This would be an arithmetic statement that they think is false, but most set theorists on Earth think is true. Or perhaps they think ZFC is inconsistent or even much weaker theories. And we know that there is in principle nothing dangerous about those beliefs, because we can prove that they are consistent relative to our theory, so we can't really prove them wrong. I think there is room for thinking that there could be genuine arithmetic disagreements.
I don’t think this is anywhere near as likely. No matter how strong their intuition was that there were no inaccessibles, after a century or so they would have come around to recognizing that this was something that needed its own axiom.
The ancients and medievals didn’t have the right attitude here, and found the counterintuitiveness of denying the parallel postulate so disturbing that they kept looking for refutations from the other Euclidean axioms for many centuries, but for a long time now people have been unperturbed when some axioms aren’t strong enough to imply another one.
Although I can’t think of any examples in the modern era of logic (1930’s onward) where it took many decades to discover an inconsistency in a system, there are examples of systems whose consistency could not be proven for a very long time, which is almost the same thing. The recent machine-verified proof of the consistency of Quine’s New Foundations system (announced by Randall Holmes a decade ago but only fully verified last year) is the best one I know of.
What’s the record for the amount of time between an axiom being suggested, and it being found to be inconsistent?
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.