[FOM] Really Large Infinitary Languages

[Alan Weir]

There are various coding 'tricks' which are used to simulate the effect of having structured entities (sets or set-like entities, sequences etc) whose components are 'proper classes' or universe-sized sets including structures of infinite size, for example in 'mouse theory' (cf.  E. Schimmerling,   “The ABC’s of mice”, Bulletin of Symbolic Logic 7, 2001 485-503.)
 
But if these methods really are tricks, as per the scare -quoting then it's not clear that this is, as it were, an honest way to do this or that one really has created infinite structures with proper classes down at the bottom. An alternative may be to appeal to the idea of indefinite extensibility and think of a really large infinitary language as an indefinitely extensible object, each precisification of which is a standard infinitary language. But personally I think the whole notion of an indefinitely extensible 'totality', 'ensemble', 'collection', or whatever euphemism one uses, is just a cover for naïve set. And that's the honest way to get really large languages, work in naïve set theory, where there is no problem about having sets containing universe-sized sets (including the universal set itself); the cost of course being substantial restrictions in the logic with one key question then arising being how much ability to work, to prove things, about those languages will one have.

Best

Alan


Professor Alan Weir
Roinn na Feallsanachd/Philosophy
Sgoil nan Daonnachdan/School of Humanities
Oilthigh Ghlaschu/University of Glasgow
GLASGOW G12 8QQ


>>Message: 1
Date: Sun, 23 Nov 2014 13:49:48 +1300
From: Guillermo Badia <guillebadia89 at gmail.com>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: [FOM] Really Large Infinitary Languages
 So, do you think it is impossible to have languages with conjunctions
of  proper classes of formulas? What about Kelley-Morse set theory?
There we have classes "representing" proper classes of proper classes.
Couldn't one build these languages in that context? Thanks very much
for your replies.