[FOM] English translation of Weyl (1910)

[Joe Shipman]

I believe you are misinterpreting Weyl’s proposal. At the time, there was no notion of the hierarchy V as we picture it now, with ordinals “vertical” and ranks “horizontal”. There is no reason a “subdomain” cannot be limited vertically as well as horizontally. Thus, Weyl is proposing the stronger axiom that there is no standard set model of ZF, in addition to there being no proper class submodel or proper inner model in our current terminology.

I’ve discussed this axiom here before. It holds in Cohen’s minimal model M, and is equivalent to the conjunction of V=L with the negation of Cohen’s axiom SM (a standard model exists). For short I call this axiom V=M, and I don’t see why it’s any less plausible than V=L. If the only sets are the ones that must exist given the ordinals and ZF, why not also say the only ordinals are the ones that must exist given ZF?

— JS

Sent from my iPhone

> On May 3, 2018, at 4:56 PM, Stephen Pollard <spollard at truman.edu> wrote:
> 
> Having failed to locate an English translation of Hermann Weyl’s 1910 habilitation lecture (“Über die Definitionen der mathematischen Grundbegriffe”), I decided to do one of my own. It is posted here: https://www.academia.edu/36559178/Definitions_of_Fundamental_Mathematical_Concepts
> 
> Corrections would be most welcome.
> 
> In one passage of particular interest, Weyl asserts that CH can be decided only if we add to Zermelo's set theory an axiom "expressing the exact opposite of Hilbert's completeness axiom.” Namely: “the domain of Zermelian things has no subdomain that (holding constant the membership relation) satisfies all of Zermelo's axioms.” This at least points in the general direction of a minimal, transitive, proper class model.
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> https://cs.nyu.edu/mailman/listinfo/fom
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20180504/41087aa1/attachment-0001.html>