FOM: categorical non-foundations; three challenges for McLarty

Kanovei kanovei at
Sun Jan 25 14:15:11 EST 1998

>Date: Sat, 24 Jan 1998 20:16:05 -0500 (EST)
>From: Stephen G Simpson <simpson at>

>Echoing Harvey's posting of 24 Jan 1998 00:13:02, I pose three very
>specific challenges for McLarty.

Let me try to simulate McLarty, just to demonstrate that 
he will not be hurted much. (Hopefully he will 
forgive my taking this liberty, and Steve this time 
will let me through to the list.) 

>Challenge 1. McLarty needs to explicitly state fully formal axioms of
>topos theory plus natural number object etc.  These axioms are to be
>compared side-by-side with the very concise yet fully formal axioms of
>set theory which were posted by Harvey in 23 Jan 1998 00:54:20.

"ML" (means: my simulation of his possible response): 
See my book (or Mac Lane or some other). Compare those at 
your own cost because in fact the setups are incomparable 
as they have different basic standpoints. 

>Challenge 2. McLarty needs to forthrightly concede that topos with
>natural number object is not sufficient for undergraduate real
>analysis, specifically the standard undergraduate material on
>calculus [...]

If this is not sufficient for you look more carefully in books on 
categories. There are toposes equivalent to your favourite 
ZFC, take them if you want. 
In general, categories is not for kids, they are for grown-ups, 
for Filds-medal seekers, so any involvement of undergrads 
and 3-year-olds with their cards on the table is not too serious. 
(I - Kanovei - am somewhat wondering why ML has not yet given 
the last argument.) 

>Challenge 3. McLarty needs to forthrightly concede that, 
>as Harvey put it,
 > there is no coherent conception of the mathematical universe that
 > underlies categorical foundations in your sense.

I agree, simply because in fact there is no unified mathematical 
universe. Indeed how you can get a uniform universe containing 
both natural numbers and geometric figures, unless you allow 
such a stupid thing as e.g. intersection of \pi and the unit 
****** (end)

Vladimir Kanovei

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