[FOM] Why not this theory be the foundational theory of mathematics?

[martdowd at aol.com]

FOM, 
 This posting of Colin's touches on vexing foundational issues, on which
he is an expert.  These go back to Russell's paradox.  One cannot consider
the "set of all sets".  However the "collection of all sets"
can be considered if one exercises appropriate care, and indeed it is
sometimes useful to do so.  This problem has vexed set theorists since
Cantor.  Various conservative extension theorems can be invoked to
justify certain "abuses", and moderate extensions of ZFC such as NBG
provide for further such, although the "standard canon" would seem to be
to avoid this in ordinary mathematics.

 
-----Original Message-----
From: Colin McLarty <colin.mclarty at case.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Sun, Mar 31, 2019 4:11 pm
Subject: Re: [FOM] Why not this theory be the foundational theory of mathematics?


> Thomas Klimpel wrote (with much else)

 
My remarks about category theory were triggered by Muller's paper
(http://philsci-archive.pitt.edu/1372/1/SetClassCat.PDF), especially
by the following passage in the middle of page 11:

> When a category-theoretician talks about **Set**, *he means all sets, i.e. all sets available in the domain of discourse of mathematics*, which has now become V+, not V  \subset V+, because old V comprises less than a thee spoon of the sets in the indefinitely expanding cosmos V+. When a category-theoretician hears there are sets available of cardinality i1, 2^i1, i1^\omega, \alpha_i1, etc., he means to include *these too* in **Set**, because these are also sets according to these stronger set-theories, not mathematical objects distinct from sets.


Of course Muller quotes no category theorists saying such things, because none does.   To the contrary, in that same paragraph, Muller quotes MacLane saying his (MacLane's) foundation does not give a category of absolutely all groups, for example.  Muller could have concluded that he misunderstood what MacLane meant by *all sets* or *all groups* in the first place.  But instead Muller says MacLane's idea is "is artificial and barks at his explicit intentions."  Notice when Muller says "explicit intentions" he means "implicit intentions," because MacLane never did utter the naive claim Muller attributes to him.

This passage gave me the impression that foundations of category
theory could turn into a fight over words. 

Indeed the discussion often has become just such a fight.   Personally I believe the best way to avoid fighting over words is to actually quote the words of people you disgree with.   If you believe category theorists make some claim, find a quote of one making it.

When I talked about category theory as too vague for a foundational
theory, I thought about the passage at the top of page 2:
> Lawvere [1966] proposed **CAT**, the category of all categories (save itself), as the domain of discourse of mathematics, and embarked on the endeavour to axiomatise **CAT** directly

Independent of whether Lawvere came up with a nice set of axioms for
**CAT** or not, this just feels too vague to me.

Here you and I agree.  Muller's sentence is entirely too vague to count as an argument for a foundation for math.  But then, it was not supposed to be such an argument.  To learn about axiomatizations of CAT you should read papers that offer them.  
best, Colin

 
 It is one thing to
formalize category theory like group theory or lattice theory. If ten
people tried this independently, at least eight would probably come up
with equivalent formalizations. But if the same ten people tried to
formalize **CAT**, they would end up with ten different non-equivalent
formalizations.

(If ten people tried to formalize set theory, none would come up with
ZFC. Some would come up with inconsistent systems like Frege, some
with second order formalizations like Zermelo, some with systems
equiconsistent with finite order arithmetic like Russell and
Whitehead, some with even weaker systems. Yes, set theory is also
vague, but ZFC is not.)


When I say "last word quantification", I try to capture an interesting
property of the full semantics of second order quantification (which
is absent from Henkin semantics). The second order induction axiom
characterizes the natural numbers up to isomorphism. Initially,
neither addition, nor multiplication or exponentiation is defined,
only successor. But as soon as a symbol and axioms for an operation
like exponentiation is added, the second order induction axiom
includes it too (as a way to define properties). Since we know pretty
well what we mean by the natural numbers, having the last word might
not be bad in this case. However, a similar game could also be played
for the real numbers (defined as a complete ordered field), but that
does not mean that we know similarly well what we mean by the real
numbers.

Thomas
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