[FOM] Really Large Infinitary Languages

Harry Deutsch hdeutsch at ilstu.edu
Sat Nov 22 20:23:33 EST 2014

On 11/22/14, 3:25 PM, Alasdair Urquhart wrote:
> I don't quite understand this question.  If you are working
> in NBG set theory, then you might write the conjunction
> of a proper class C of propositions as the ordered pair
> < &,C >, let's say.
> But then C would be a member of a class, which is impossible
> in NBG.  So, the formulation of the question is not
> clear to me.
> On Sat, 22 Nov 2014, Guillermo Badia wrote:
>> Max Dickmann's book "Large Infinitary Languages" contains a 
>> discussion of
>>> proper class sized Infinitary languages.
>>> -- John Bell
>> Thanks a lot. In what section exactly? I haven't find it. Please note
>> that I don't mean languages with a proper class of formulas, but
>> languages with conjunctions and quantifications of proper class size.
>> Cheers
>> Guillermo
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Alasdair, one can use a "flat" pairing operation rather than the 
Kuratowski pair--which raises the types of the components.  Robinson 
(JSL, 1945) apparently gave the first such definition, but Quine also 
did so later. (See Drake: Set Theory). So there is no problem having 
finite sequences with proper classes as components.  I don't know if 
such flat sequences can be extended to the infinite.  But I do not see 
any immediate reason that can't happen. In any case, the question was 
whether there can be proper class sized infinite conjunctions or 
disjunctions. I don't know the answer to that.

Harry Deutsch

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