[FOM] Formalization Thesis

T.Forster@dpmms.cam.ac.uk T.Forster at dpmms.cam.ac.uk
Sat Dec 29 06:45:27 EST 2007

Tim, Thank you for your interesting post. It has unleashed a storm of 
discussion, as i knew it would - none of which have i had time to read - 
tho' i am looking forward to it. I am risking contributing this post 
without having read the rest of the correspondence beco's i suspect that 
the point i am about to make will be made by no-one else except possibly 
Randall Holmes, and he seems to be off-line.

     I am thinking about the attitude to the paradoxes taken by ZF and its 
congenors. The basic message it brings is that there is a simple uniform 
explanation of the paradoxes, and that is the error of thinking that the 
problematic objects are sets. (As you know, i am a student of a set theory 
that doesn't take this point of view, so of course i would be saying all 
this wouldn't i!) We should remember that altho' the non-sethood of some of 
these problematic collections is a theorem of pure logic (LPC), the 
non-sethood of - for example - the universe - is not. Perhaps this 
difference matters..?

     Chow's principle could be false if there are genuine mathematical 
facts about some of these large dodgy collections (V for example) that do 
not lend themselves to representation as facts about wellfounded sets. I do 
not have any compelling examples of such mathematical assertions, but i am 
alive to the possibility that there might be some. By setting its face 
against taking these entities seriously ZF & co are betting on there *not* 
being many such mathematics. There may, indeed, not be any such 
mathematics, but it's by no means an open-and-shut case. It would be if the 
nonexistence of all the naughty objects were provable in predicate calculus 
but it isn't.

       Happy new Year

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