Repeating myself on RVM

[Joe Shipman]

I strongly disagree with your (1) because you called the meaningfulness of P(N) “the” foundational debate about CH. It may be “a” foundational debate about CH, but you are taking an awful lot for granted about the views of an awfully large number of mathematicians.

I also disagree with your (3), both because it’s not clear how you can be confident of this, and because it’s not clear why axioms have to be “obvious” initially, rather than becoming that way over time as they are explored (AC is much more “obvious” now than it used to be 100 years ago, and the axiom of (a completed) infinity is much more obvious than it used to be 200 years ago, and the axiom of Replacement was certainly not obvious enough for Zermelo to think of it, etc.).

To answer your (4): I am trying to come up with axioms FOR MATHEMATICS, not for set theory. Although Set Theory has been extremely successful in providing a foundation for mathematics, it so happens that certain more mathematical and less set-theoretic axioms (such as the extensibility of Lebesgue measure, or the determinacy of projective sets) have important CONSEQUENCES in set theory, and therefore one may start with the mathematical version of the axiom rather than a set-theoretic consequence. I could have said “a real-valued measurable cardinal exists”, or even “a measurable cardinal exists”, if I cared more about arithmetic or about abstract set theory rather than about measure theory, but I think the version I used is the most “plausible” and in the past might have even been considered “obvious”.

— JS

Sent from my iPhone

> On Aug 3, 2020, at 9:11 PM, Haim Gaifman <hg17 at columbia.edu> wrote:
> 
> 
> Here are some points concerning the question posed by Joe Shipman:
> 
> (1) The very notion of "being too restrictive" is misleading. At best, the argument in which it is used is question begging.
> The foundational debate about CH is not about the true cardinality of P(N)  (where P(N) = power set of N, and N=set of natural numbers)
> but about the very meaningfulness of the notion of a set that contains all subsets of N.
> 
> We know  of subsets of N because we can construct them using particular definitions. We can also define families of subsets, by using
> definitions the depend on parameters ranging over N. But the notion of the absolute totality of all subsets of N, 
> subsets whatsoever,  irrespective of a way of getting them by any definition or construction,  is highly suspicious.
> If nonetheless you accept this notion, then of course, V=L might appear restrictive. But the debate is about
> this very legitimacy of the very notion of P(N).
> 
> (2) V=L legitimizes the notion of an arbitrary set, if you assume an absolute notion of well-ordering.
> 
> (3) It appears, by now, that attempts to answer the question about the cardinality of P(N), by assuming some additional set-theoretic axiom,
> are doomed to fail, because  any such  axiom  will not be "obvious".
> 
> (4) Why should we accept the axiom about extending Lebesgue measures, as a set theoretic axiom?
> 
> Haim Gaifman
> 
> On 8/2/2020 11:59 PM, Joe Shipman wrote:
>> I read a lot of papers which talk about the unsatisfactoriness of “new axioms” for non-absolute statements like CH. It seems clear that most people working in the area don’t like V=L and related axioms because they are too restrictive about that sets may exist, and feel like prospects for settling CH are dim.
>> 
>> But I have a still never heard a satisfactory explanation of what is wrong with the axiom that Lebesgue measure can be extended to all sets of reals in a way that remains countably additive (though no longer translation-invariant).
>> 
>> What is an example of an independent-of-ZFC statement anyone cares about that this axiom does NOT decide (apart from propositions implying the consistency of cardinals larger than “measurable“)?
>> 
>> — JS
>> 
>> Sent from my iPhone
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