On the axiom of constructibility and Maddy’s conception of restrictive theories, Logic Workshop, February 2013

This is a talk for the CUNY Logic Workshop on February 15, 2013.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Set-theorists often argue against the axiom of constructibility $V=L$ on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible.  Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory.

In this talk, I shall criticize Maddy’s specific proposal.  For example, it turns out that the fairly-interpreted-in relation on theories is not transitive, and similarly the maximizes-over and strongly-maximizes-over relations are not transitive.  Further, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s proposal, although this is not what she had desired.

Ultimately, I argue that the $V\neq L$ via maximize position loses its force on a multiverse conception of set theory, in light of the classical facts that models of set theory can generally be extended to (taller) models of $V=L$.  In particular, every countable model of set theory is a transitive set inside a model of $V=L$.  I shall conclude the talk by explaining various senses in which $V=L$ remains compatible with strength in set theory.

3 thoughts on “On the axiom of constructibility and Maddy’s conception of restrictive theories, Logic Workshop, February 2013

    • Asaf, you flatter me! We haven’t been making our workshop available this way, but we’ve said we might want to do this in the future, and perhaps it is time to look into it. Does anyone have some suggestions about how the best way to go about it? I know that Francois recently had done this for his talk at Dartmouth, and I think also that Peter has arranged this for some of his seminars. Any advice?

      • If you only want to record, any modern HD camcorder will probably do. Flipcam-like camcorders the Samsung W300 start at ~$100, plus an SD-card, maybe another 30.

        To push things further, you could consider live-broadcasting. This used to be a pain (I remember broadcasting Hugh Woodin’s Ziwet lecture via justin.tv was hellish…)

        But these days you have google+ hangouts which make it terribly easy — easy enough to have a seminar in Michigan with visitors from Prague and Kobe simultaneously. For this you should probably invest in an HD webcam. Not that hangouts allow HD (yet?), but HD cams are usually better in quality than built-in laptop webcams (especially the microphone which is very importnat). Also, they start ~$50 (there’s really only Logitech and Microsoft these days).

        The good thing about hangouts is that you can make them public (which allows anybody to watch them as a youtube stream, in addition to the 9 people who can join them actively) and which will record them as well. You can have slides in parallel by sharing a screen etc. If you’ve never tried hangouts, you should test it.

        The bad thing is that the current resolution of hangouts makes it hard to get a black/whiteboard image across. Blackboard should be really clean (as in freshly cleaned with water) to get enough contrast. Whiteboards need big markers…

        Recently, the TCS community also started using hangouts for global seminars.

        Anyway, Francois will have more practical advice since he continued doing this academically — I just use it for fun these days 😉

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