[FOM] Epistemology for new axioms

[Joe Shipman]

I’d like to clarify my previous post with a more detailed example.

I can imagine, and have described here before, good arguments that “a real valued countably additive measure exists on the continuum” is true or should be an axiom. It’s known that this is equiconsistent with a measurable cardinal, so it can’t be proven consistent in ZFC, but it has the consequence that CH fails badly. Thus one could ask if this contradicts my proposal, because ~CH is an axiom ZFC shows is relatively consistent and independent that could indeed achieve a near-consensus status.

It does not, though, because the justification for ~CH is that it is a consequence of a stronger axiom for which there are good arguments.  I don’t see any prospect of persuading people that CH or ~CH is true without also arguing for statements ZFC can’t prove are safe.

In 1990 I showed that physicists had proposed theories which assumed CH in order to make sense, which is the kind of “evidence” for new axioms Godel once envisioned, and a reasonable axiom candidate was the failure of “strong Fubini axioms” that iterated integrals on product spaces were always equal when they existed even for nonmeasurable functions (with the technical condition that the functions were either defined on compact spaces or bounded below). But although these anti-Fubini axioms were relatively consistent because they followed from CH, and I showed they were independent and thus needed to be assumed, the physics was quite unsatisfactory so I did not regard this as a successful argument for a new axiom.

I currently believe that any arguments from science won’t suffice to establish plausibility for non-absolute statements, but I’d love to be shown wrong on this. I’d also love to see a mathematically interesting absolute statement, especially an arithmetical one, that ZFC shows is consistent and independent.

— JS

Sent from my iPhone

> On Sep 3, 2019, at 2:53 PM, Joe Shipman <joeshipman at aol.com> wrote:
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> I didn’t get any responses to my queries about absoluteness, so I challenge you to argue against the following: 
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> ***There is no proposition of any mathematical interest that is known (by ZFC proof) to be relatively consistent with and independent of ZFC, such that mathematicians will eventually have a consensus view that it should be considered as a fundamental axiom with which ZFC should be augmented.***
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> Here “a consensus view” means comparable to the view that Zermelo set theory should be augmented with the Replacement Axiom to give ZF. In my opinion, statements like CH or V=L or their negations will never be settled because there is no way for a believer to persuade a denier, they will never lead to consequences for which any kind of empirical support can be given such as numerical evidence or necessity for science. 
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> (This opinion isn’t as strong as the proposition I am challenging you to argue against, maybe there are interesting but more concrete statements that ZFC can show are consistent and independent even though no one suggested any when I asked last month.)
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> This is to be contrasted with axioms that can’t be shown within ZFC to be consistent and independent, like Con(ZF); I don’t claim there will be permanent disagreement about statements like that.
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> In fact, for arithmetical statements, I claim the opposite: although there may always be arithmetical statements which some mathematicians will claim to have proven while others will not accept the proofs, there will never be an arithmetical statement such that two groups of mathematicians will permanently disagree about it, with one group claiming it has been proven and the other claiming it has been disproven.
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> “Permanent disagreement” is the kind of thing you get in physics when people disagree about things like “interpretations of quantum mechanics” in a way that experiment can’t settle even in principle; I approve of the professional physicists who dismiss such arguments as “religious disputes”, and wonder how much effort is wasted on the mathematical equivalent.
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> I’ve proposed two epistemological classes here: one includes non-absolute statements like CH, where I expect permanent disagreement about whether to prefer X or ~X, and one includes all arithmetical statements, where I deny there can be permanent disagreement.
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> In between, we have statements which can be set-theoretically absolute but where it’s open whether some kind of empirical support can be found to favor one side of the question over the other.  I think it’s a very interesting project what kind of empirical support there can be for non-arithmetical statements. For example, compare the problems of persuading non-believers that 
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> (a) measurable cardinals exist
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> (b) real-valued measurable cardinals exist
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> (c) measurable cardinals are consistent
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> Don’t interpret my stance as support for the “Multiverse” view of set theory proposed by Hamkins and others. One can regard CH as having a real truth value even if it is not knowable by human beings. It’s an inverse of Kronecker’s famous dictum: “Man can know the integers, but higher infinities are only knowable by God.”
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> — JS
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