Monday, 15 July 2013

SGM: Sui Generis Mathematics

The Cantorian transfinite cumulative hierarchy $V$ of pure wellfounded sets has an amazing property:
Every mathematical object that anyone has ever thought of can be "represented/modelled" in $V$.
That is: pairs, relations, functions, tuples, sequences, ordinals, cardinals, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, infinitesimals, etc., S_n, graphs, groups, vector spaces, topological spaces, manifolds, fibre bundles, etc., etc., can all be represented in $V$. In fact, most of them live happily inside the rank $V_{\omega + n}$, with $n$ around $3$.

One might object:
Ah, but what about proper classes?
Well, ok, these are structures that are just as large as $V$ itself.

A much more serious objection (based on Benacerraf's famous 1965 article "What Numbers Could Not Be"):
"Represented" yes, but that doesn't mean that, e.g., the real number $\pi$, or the group SU(3), is a set. Furthermore, extensionally inequivalent representations exist. E.g., there are two distinct sets $X$ and $Y$ both of which can represent/model the natural numbers. And this phenomenon of non-uniqueness is pervasive. In a sense, there are lots and lots are arbitrary choices. It doesn't in fact ever matter which "choice" we make, unless we ask a dumb question.
(There is a technical reason for it not ever mattering: these reductions can be formulated as definitional extensions of $ZFC$ and definitional extensions are conservative.)

These "arbitrary choices" are, I believe, rather like gauge choices. We can choose some set $X$ to represent $\mathbb{N}$, or $\mathbb{R}$, etc., and in many inequivalent ways. But perhaps we would like to have natural numbers and real numbers living in the mathematical universe quite independently of any reduction. They would be sui generis mathematical objects. Sui generis mathematical objects will always be representable in $V$. But they live alongside $V$, rather than inside $V$.

How might one give a theory of these objects which did not require some arbitrary choice for the reduction to sets? What would a "gauge-invariant" theory of sui generis mathematical objects---pairs, relations, functions, natural numbers, ordinals, etc.--- look like?

I think the answer is this:
Extend set theory with abstraction principles for sui generis mathematical objects.
The kind of set/class theory one might start with is $\mathsf{ASC}$, "atoms, sets, and classes", as described in this M-Phi post, along with requisite set existence axioms. If one adds the usual set existence axioms, one gets a theory simlar to $\mathsf{MKA}$, Morse-Kelley set (class) theory with atoms.

For the sui generis entities, one can then consider quite a variety of abstraction principles for sui generis mathematical entities. Here are three:
(T-Abs) $\sigma_n(x_1, \dots, x_n) = \sigma_n(y_1,\dots, y_n) \leftrightarrow x_1 = y_1 \& \dots \& x_n = y_n$.
(N-Abs) $|X| = |Y| \leftrightarrow X \sim Y$.
(E-Abs) $Equiv_{\phi} \to ([x]_{\phi} = [y]_{\phi} \leftrightarrow \phi(x,y))$.
where $\phi = \phi(x,y)$ is a formula with two free variables, and $Equiv_{\phi}$ says that $\phi$ expresses an equivalence relation.

Intuitively:
• (T-Abs) is an abstraction principle yielding tuples;
• (N-Abs) is an abstraction principle yielding cardinal numbers
• (E-Abs) is an abstraction principle yielding "equivalence types" (e.g., isomorphism types).
Define $\mathsf{SGM}$ (for "Sui Generis Mathematics") to be the result of adding these abstraction principles to $\mathsf{MKA}$:
$\mathsf{SGM} := \mathsf{MKA}$ + (T-Abs) + (N-Abs) + (E-Abs).
Then I believe---but I'm also not entirely sure---that the following conjecture is correct:
$\mathsf{SGM}$ is a conservative extension of $\mathsf{MKA}$.
For given a model for $\mathsf{MKA}$, then we can interpret each abstraction principle, in the fairly obvious way, to come out true. For the abstractions with tuples and equivalences, we know how to do this. For the abstraction given by (N-Abs), i.e., Hume's Principle, then the abstracta $|X|$ for set sized classes $X$ are interpreted as cardinals, in the usual way (i.e., initial ordinals). Any proper class sized $X$ has the same cardinality, $=|V|$: call this $\infty$. This can presumably be interpreted as some arbitrary object that is not an initial ordinal.

In more detail, $\mathsf{SGM}$ implies the existence of anti-zero, $\infty$, i.e.,
$\mathsf{SGM} \vdash \infty = |V|$
Assuming there are atoms, then this unique very "large" entity could be interpreted as some atom. In fact, I am inclined to add as an axiom:
All sui generis abstracta are atoms.
But this would then (in the intended model, as it were) force the collection $At$ of atoms to form a proper class sized collection. And the cardinality $|At|$ of the collection $At$ of atoms would be $\infty$, and also an element of the collection of atoms. I.e.,
$\infty \in At$.
But I think this is all ok.

1. Dennis Kavlakoglu15 July 2013 at 19:00

"Extend set theory with abstraction principles for sui generis mathematical objects."

But doesn't this give rise to another charge of arbitrariness? Why choose set theory as the base theory to be extended? Why not embed the abstraction principles in a second-order logic (a la Neo-Fregeanism) this way you can at least avoid the charge that you're privileging one mathematical theory over another?

2. Thanks, Dennis

Actually, the theory itself is already sort-of-second order, because it starts with a theory of classes & atoms, with impredicative comprehension (I call this ASC). Sets are defined to be classes that are members.

Usually, abstractionists/neo-logicists have tried to get sets from SOL using a refined abstraction principle (a modification of Frege's BLV); but they haven't been successful, because to get something like the cumulative hierarchy of sets out of SOL + abstraction for sets, one has to make complicated structural assumptions built into the abstraction principle, which are more or less equivalent to Zermelo's axioms.

So, instead, I don't try and reduce sets to extensions of classes, but leave them as they are. So, I can treat sets themselves as sui generis mathematical objects (given by set existence axioms, a la Zermelo); and then all the other abstracta arise from appropriate abstraction principles.

The point isn't to eliminate sets; the point is to extend the set-theoretical universe with sui generis abstracta, so that arbitrary reductions are not required (except to check consistency/conservation).

Cheers,

Jeff

1. Dennis Kavlakoglu15 July 2013 at 19:44

Does the usage of set theory as the base theory to be extended commit you to some view of set theory as "foundational"? Clearly not in the sense of being a reduction base, but it seems to me there still might be some priority claim you're committed to in virtue of utilizing set theory as the base theory. This is only a hazy worry though, and perhaps something that can be easily skirted.

I am actually rather sympathetic to this sort of anti-reductionist project. I look forward to future posts (and/or publications) on this topic.

3. Thanks, Dennis

Glad to hear you're interested! Yes, anti-reductionism is part of the motivation. But I think there's no additional need to eliminate sets. The foundational/base part is really the system $\mathsf{ASC}$ ("atoms, sets and classes"), here,

http://m-phi.blogspot.co.uk/2013/07/asc-atoms-sets-and-classes.html

But this theory $\mathsf{ASC}$ (of classes) is very weak, and doesn't imply the existence of any sets. To get the existence of sets, one has to add specific set-existence axioms saying things like "the class $\{x \mid x \neq x\}$ is a set", etc.

Everything else---i.e., the sui generis abstracta---is then added by abstraction principles. The difference with recent neo-logicism is that one has given up on the idea of generating sets from abstraction principles. As sui generis abstracta, they needn't be reduced to sets; but they can be (to check consistency, say); and the reduction to some set is like a "gauge choice".

Cheers,

Jeff