FOM: Axioms:reply and two corrections

Colin Mclarty cxm7 at
Sun Feb 1 02:40:15 EST 1998

	Charles Silver said it seems that some people on each
side of the SET/TOP debate regard simple formal axioms as a
criterion for fom, and he is probably right. But I do not
regard it as a criterion. I think simplicity of axioms is one
nice goal. I like the axiom systems Harvey gave, and freely
agree they have an aesthetic unity mine lack. On the other hand
I claim that 15 is substantially less than infinite, and in this
sense my system is simpler than his. Neither fact seems to me
decisive for fom. I posted my axioms because Harvey and Steve 
Simpson asked me to. They are certainly not how I see the 

	 I do think any candidate for fom must be first order 
expressible, I do not think the particulars of the expression 
weigh very heavily on the question of whether something can 
be fom.

	Niel Tennant and Soren Riis both pointed out a typo:
>> 2.  (f,g,h)(there exist A,B,C) [ (f:A-->B and g:B-->C) iff  
>>        (for some h)M(g,f;h)]

The "h" in the first quantifier (which, technically, is idle
anyway) should be deleted. And the openning square bracket should
go before the existential quantifier.

	So the first line reads

	(f,g) [ (there exist A,B,C)(f:A-->B & g:(B-->C)  iff

Soren Riis made me notice my subobject classifier axiom only says
subobjects have classifying arrows, where it should also say
every arrow to *W* classifies a subobject. This makes a comprehension
axiom. Here I quote the original and follow it with the addition:

> 12.  (f,A,B) [If (m(f) & f:A-->B) then (there exists a unique 
>        such that (h,k,j)[If (M(k,t;j) & M(u,h;j) then 
>                (there exists a unique v such that M(f,v;h))]] 

	& (u,B) [ If u:B-->*W* then (there exist f,A)(f:A-->B and
		(h,j,k)[If (M(k,t;j) & M(u,h;j) then
		(there exists a unique v such that M(f,v;h)]]

I apologize for any confusion this caused. Soren developed a rather
good sketch of a model for the axioms as I gave them, clearly 
inadequate for analysis as it had far too few functions between
the various powers of the natural numbers N. 


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