Repeating myself on RVM

[Haim Gaifman]

Dear Joe Simpson

For the sake of convenience let me enumerate the points of my answer.
(I) I certainly stated an anti-realist position with regard to set 
theory. In particular, I do not think that the question "what is the 
cardinality of the continuum"
has an objective answer. My position is rather similar to the position 
of  Shelah in his survey paper from 2000.
You say that you strongly disagree with me because I am  "taking an 
awful lot for granted about the views of an awfully large number of 
mathematicians". But I made no claim about the
views of other mathematicians. My argument concerns a certain question, 
not the statistics of views in the mathematical community.

(II) You said that most people working in the area found V=L "too 
restrictive about that sets may exist". This describes the position 
advocated in Penelope Maddy's  book from 1900 "Realism in mathematics"
which was quite fashionable among philosophers of Math. Things have 
changed since then. In a paper by Joel Hamkins published in 2014, he 
proves that Maddy's intuitive criteria for having "as many subsets of a 
given set as possible",
cannot be stated in a rigorous form. In any case, the rejection of V=L 
depends on the acceptance of the very notion of the "set of /all/ 
subsets of a given set", and it is this very notion that is under debate.

(III) The Lebesgue-measure axiom considered by you, is not stated in 
set-theoretic terms.
It is however is a claim about the truth of a certain set-theoretic 
statement in the universe V_(*ω* +/k/)   (the set of all sets of rank 
≤*ω* +/k/) where// /k /is some small integer (I believe /k/ = 3 would do).
Therefore it amounts de facto to a set-theoretical axiom.

(IV) You did indicate an answer to my last question, the kind of answer 
which I agree with: This axiom makes for a "smoother" or "nicer" measure 
theory, or it gives us insights into it.
Of course the question of  a relative consistency proof remains 
important. In the same way, algebraists   accepted the axiom of choice, 
once they realized that it is necessary for
  getting  a transcendency basis and define the transcendency rank in 
the theory of fields. But this does /not/ show that  the axiom of choice 
is "true". It only shows that it is required if
we want to have a field theory with such and such properties. This is my 
position, which is more or less the same as that of Shelah. I think that 
it will fit nicely your project.
I should not worry about philosophical objections that arise from a 
realistic conception of set theory. (The situation is different when it 
comes to more elementary parts of mathematics, such
as number theory).
Good luck.
Best HG


On 8/3/20 9:46 PM, Joe Shipman wrote:
> 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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