[FOM] Epistemology for new axioms

[Joe Shipman]

I didn’t get any responses to my queries about absoluteness, so I challenge you to argue against the following: 



***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.***



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. 



(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.)



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.



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.



“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.



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.



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 

(a) measurable cardinals exist

(b) real-valued measurable cardinals exist

(c) measurable cardinals are consistent



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.”



— JS

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