friedman at math.ohio-state.edu
Fri Jan 16 23:06:59 EST 2004
On 1/14/04 1:14 PM, "Thomas Forster" <T.Forster at dpmms.cam.ac.uk> wrote:
> An essay question:
> Are the finite ordinals the same mathematical objects as the finite
> cardinals? Give reasons... [\aleph_0 marks]
This is a particular case of an old and heavily discussed issue in f.o.m. It
is all too easy to say mundane things about this matter, merely reproducing
what has been interminably discussed in the literature and elsewhere for
So let me try to suggest something nonmundane, or arguably nonmundane, about
this. Three main approaches to f.o.m., relevant to this matter, come to
1. Make the ontology of mathematics as minimal as possible. Take the
position - at least theoretically - that every mathematical object has an
immutable objective reality independently of context. (I.e., avoid such
approaches as "categorical foundations" which, in its most ambitious forms,
view mathematical objects as meaningful only in the context of structures,
etc.) All mathematical objects are to be forced to be one of the primitive
objects. Define appropriate notions of isomorphism between structures, and
prove that simple conditions on structures imply the existence of
isomorphisms (often unique isomorphisms). NOTE: In this approach, systems,
like all mathematical objects, must themselves be one of the primitive
objects, and the simple conditions placed on structures that imply the
existence of isomorphisms (often unique) correspond directly to the use of
these systems in actual mathematics, and their motivation for being
introduced in mathematics.
2. Make the ontology of mathematics as diverse as at all reasonable. Still
take the position that every mathematical object has an immutable objective
reality independently of context. The idea here is that just about all
fundamental mathematical entities are not sets, but rather urelements. So we
have a huge number of extra unary predicates (and associated multivariate
predicates and operations). One can go to an extreme, where even positive
integers are treated not as a kind of integer, but rather as another kind of
urelemente. I think that such a wild extreme can be distinguished from more
reasonable control of the proliferation of notions.
3. Take the idea that mathematical structures are paramount, and individuals
are not. I have never seen this worked out coherently and productively.
Where do the structures come from, and where do the objects in the
structures come from? Most working mathematicians like this approach at the
working level, but readily see that a fully autonomous development is not
only unnecessary, but cumbersome to the point of it being quite unclear just
how to proceed. So they readily accept, say, the definition of a category as
a set of objects together with another set of objects called arrows,
etcetera. So they are happy with a set theoretic underpinning of
mathematics, and don't want to be bothered with foundational issues,
assuming, confidently, that they have been resolved - or at least resolved
sufficiently clearly for their purposes.
4. Other approaches, that have not been considered systematically to my
knowledge, and which I won't go into until I have something productive to
say about them.
Obviously 1 is the most common approach to f.o.m., and it serves us very
well. There is something particularly attractive - and simplifying!! - about
immutable objects of objective character, with one nonlogical relation.
I.e., sets and membership (in addition to equality, connectives,
I have never seen a fully developed treatment of approach 2, which should
include at least working criteria for when we should introduce a new unary
relation symbol (new kind of urelement) and when we should not.
On the FOM, I had previously discussed a limited form of 2 by taking the
ordered ring of real numbers as primitive, and rewriting the axioms of ZFC
incorporating this new primitive. I could have instead used the ordered
group of real numbers, with a constant for 1, and prove that there is
exactly one multiplication operation that makes this into an ordered ring (I
think I previously mentioned this on the FOM).
Here is what I wrote in
>We have variables of only one sort, but with the following 7
>nonlogical symbols (in addition to the logical symbols not, and, or,
>implies, iff, forall, therexists, =).
>Sets. (Unary predicate symbol).
>Membership. (Binary relation symbol).
>Ordered pairing (Binary function symbol).
>Real numbers. (Unary predicate symbol).
>0,1. (Constant symbols).
> <. (Binary relation symbol for ordering of reals).
>+. (Ternary relation symbol for addition on reals).
>1. Everything is exactly one of: a set, an ordered pair, or a real number,
>2. Only sets can have an element.
>3. If two sets have the same elements then they are equal.
>4. <x,y> = <z,w> iff (x = z and y = x).
>5. 0,1 are distinct real numbers.
>6. +(x,y,z) implies x,y,z are reals.
>7. x < y implies x,y are reals.
>8. Usual axioms that reals are an ordered group with 0,1,+,<.
>9. Every nonempty set of reals bounded above has a least upper bound.
>10. The set of all reals numbers exists.
>11. Pairing, union, power set, separation, replacement, foundation, choice.
>Rationals, integers, natural numbers, are all defined as certain real
>numbers. Functions are sets of ordered pairs.
>One proves that every sentence is provably equivalent to a sentence
>that mentions only epsilon.
Note that I didn't treat ordered pairs as urelements. If one wishes to adopt
approach 2, one would want finite tuples to be urelements, with appropriate
axioms. For that matter, it seems that one would also want infinite
sequences as urelements, also with appropriate axioms.
So the question is: just what should a fully formalized treatment of 2 look
like? It seems that any two competent people would create at least somewhat
different fully formalized treatments of 2. *Whereas, any two fully
competent people would create the same fully formalized treatments of 1 (at
least after some well known further investigations).*
How do we establish the robustness of the formalized treatments of 2? Of
course, we at least expect that all such systems based on ZFC would be
conservative extensions of ZFC. But we would want much sharper relationships
between different full formalizations of 2.
In the case at hand, we would treat ordinals as urelements, and cardinals
also as urelements. We certainly would not have any axioms that imply that
any ordinals are cardinals. Do we want to add axioms that imply that no
ordinal is a cardinal? That depends on just what general approach we want to
take towards a full formalization of 2.
Thus we may want to focus only on various full formalizations of 2 such that
#any isomorphism between the purely set theoretic parts of any two models
extends to an isomorphism between the two models.#
Under this idea, the natural thing would be to add an axiom that states that
no ordinal is a cardinal.
Let me close by saying that I am dubious about any connection between the
questions like those raised by Forster and Platonism.
Harvey M. Friedman
More information about the FOM