[FOM] Really Large Infinitary Languages

martdowd at aol.com martdowd at aol.com
Sun Nov 23 09:51:55 EST 2014


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.



Clearly it is possible to do this.  If a formula is a proper class then a collection of formulas is a type 2 object (0=set, 1=proper class).  ZFC can be extended to finite height types (this system was mentioned in Cohen's 1965 book), and in fact to types of any ordinal height.  Categories can be defined for ordinal heights, although those of finite height are netter behaved.  Co-completeness for some such categories is proved in
 Higher type categories.
 Math.\ Logic Quart.\ 39 (1993), no.\ 2, 251--254.
Undoubtedly basic facts about formulas of proper class size can readily be proved.  These can also be investigated as formulas of size kappa in V_kappa where kappa is an inaccessible cardinal.

- Martin Dowd

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