Re: "Choosing" the choice relation

Adam Burley wrote:

Adam: in general people appreciate it if you don't top-post, but rather
interleave your response with the quoted text so that it is given in
context. It's easier to read if one wants to consult the references.

At this point it is fitting to make a disclaimer: I am NOT a logician. I
have taken some logic classes, thought about the subject, and this is all
just my opinion. Though I do believe it is based on fact.

Ryan,

Thank you for your reply.

I do not understand why making a single choice is any more or less easy
than making an infinite number of choices. The problem is, that these
choices are arbitrary. As specified in the Wikipedia article,
deterministic infinite choice is not a problem. However on the other hand,
I think that non-deterministic finite choice should be a problem. The
article gives the example of the real open interval (0,1); well certainly
we can choose a real number x from this set, say x = 0.5. This is actually
deterministic. However if we have a general set X as in the countable
subset proof, then we cannot deterministically choose a single element.

You say "you don't worry, when picking an arbitrary element of a single
set, that it isn't deterministic". I do worry! I would say in formal
language that I am letting x equal the unique x with (x,X) \in C. This
then becomes a deterministic choice (once we have chosen the choice
relation non-deterministically, which is where my problem lies). However
in general terms I would omit that formality and say "pick any one", but
this is merely an abbreviation.

Let me elaborate on your position: you believe that, given any set X, to
choose an element of X is to employ a decision procedure that specifies
such an element. You also believe that this decision procedure should be
deterministically chosen from among the various options available.
Logically, you should insist that this meta-choice also be deterministic,
and so on. If determinism is what you crave, this approach is clearly
inadequate. How do you know, for example, that when you choose an element
from (0,1) you should take the midpoint rather than, say, 1/3? Or any
other point that can be specified by some algebraic formula? Not that
algebra is the only possible structure on (0,1) that could form the basis
for such a procedure. How would you choose an element of (0,1) u (2, 3)?
Would you allow choosing numbers which are merely uniquely defined by a
particular property? Chaitin's constant (see Wikipedia) comes to mind
here.

But I'm giving you a hard time. Suppose more generously that you mean that
when you choose an element, you want a way to "name" it other than simply
as being an element of X. They are not all the same, after all, right?
Since a set consists of its elements, and since you can choose any element,
they should all have names. Your names will be given in words, no doubt,
because if you can't speak the name you haven't really given it, have you?
(The mere existence of a name you can't provide is the same problem as
choosing an element of a set nondeterministically.) There are only
countably many such names, and of them only a proper subset will be valid,
meaning sensible in the language you understand. How to provide a valid
name must be described finitely on the basis of finitely many elementary
terms and operations. I will ignore the problem of determining which terms
and operations to use at any time, which would otherwise require its own
scheme much like this...

I'm basically saying that if you want everything to be deterministic then
you will have to make everything computable. You can do set theory that
way, I've never looked into it, and I don't care to. I've heard that
Norman Wildberger is a strong proponent of this approach, to pick a name I
remember (rather than, necessarily, the best name). What's distasteful
about it for me is that it eliminates all variety or abstraction and
requires that we work in a single model that doesn't even have all the
richness of my own mental image of it (see again Chaitin's constant). This
goes even for somewhat weaker restrictions like Godel's axiom "V = L" that
every set is "constructible", where instead of things being computable,
they are merely definable (one could argue that this is what I was
describing above, rather than computability). And I feel that "V = L" is
already too abstract for your tastes, since the possible definitions are
infinite in number and it is therefore not possible, even in principle, to
have a handle on them all at once. There would always be things that have
names but are unknown to you. If you make a choice now, at time T, you
would be using a conception of a set based on incomplete information, but
since you deny the existence of elements you can't specify, it would seem
that if I told you at time T + 1 of a new name, your set would change....

But this is a personal preference. Even if I admit the existence of
undefinable elements I am free to work only with the definable ones. It
may even be beneficial. One thing I learn doing math is that sometimes,
even if I am only interested in something very specific, I may need to
accept a huge and incomprehensible universe to give it context and meaning,
even if I never see most things in that universe. I don't sweat it;
they're just there to hold down the fort.

The main problem I would have is when making a definition. For example,
consider Godel's axiom of infinity (this is different from the ZFC axiom
of infinity, as it does not directly assert the existence of an inductive
set. Godel's axiom basically states that there exists a set A such that if
x is in A, then there is a superset of x in A). Now, I would want to use
this to make some definitions. This is crucial, because in NBG, the only
set which is required to exist is this infinite set. So for example you
would define the empty set 0 as the intersection of A and its complement.
However, if there is no deterministic way to choose A, then we cannot make
this definition.

We may say that this is possible when the definition forms a unique set
regardless of the choice of A. However again, I would see there being some
formal language underlying the statement "let 0 be the unique class formed
by taking the intersection of A and its complement, where A is any set
with the property that if x is in A then there is a superset of x in A".
By saying this, I would understand "let 0 = A intersection A^c, where A is
the unique element with (A, B) \in C, where B is the class of all sets A
such that if x is in A then there is a superset of x in A". Again, this
requires a deterministic choice of C.

I think this is the point in time where you really need to make a
distinction between model theory and pure logic. Logic can only talk, and
if you have a model you may even be lucky enough to have things that it's
talking about, but symbolic logic alone does not create a context for its
assertions. Suppose, for example, that you had a model for NBG (this would
mean that proper classes are "sets", but not internal sets. This is a
point of logic which I will never be satisfied with and which we are not
going to discuss in this thread). Then you would be assured of the
existence of your infinite set A, but of course not of its identity. To
rectify this, add a constant, call it A, to your language and replace
Godel's axiom "There exists an infinite set" with the more specific one "A
is an infinite set". A is required to have an interpretation in your model
and that specifies it uniquely. Indeterminacy vanishes. Basically I'm
saying: once you are in the situation of having the axioms of NBG
interpreted as actual sets, existential statements like these are no longer
abstract: there is, actually, something specific that satisfies them. The
theory doesn't tell you what it is; the model does. But we work in the
model, right?

You can even do this for more complicated statements involving nested
quantifiers. Suppose we have some statement "For all x, there exists y
such that P(x,y)". We can get rid of that existential by adding a function
symbol to the language, y(.); then the axiom can be replaced by "For all x,
P(x, y(x))". The function is required to have an interpretation in the
model and that makes it deterministic.

Note that I'm not claiming that the existential quantifier is superfluous:
for example, "There exists x such that P(x)" is equivalent to "It's not
true that for all x, P(x) is not true". So if I tried to model existential
quantifers by using deterministic constant symbols, I would end up
concluding that it is in fact DIFFERENT that "P(x) is not, for all x,
false" and "P(x) is true for some x". That would be intuitionistic logic,
I believe, which again I have nothing to do with. But perhaps this is
where YOU are headed.

On another note, you are right to say that the choice function C is a
proper class. This means we cannot, for example, take a class of all
choice functions and then pick one from that using some finite choice. If
this were not the case, then perhaps an infinite chain of axioms could be
created, terminating in the axiom of choice, and each picking the next's
choice relation (ironically, one of the main triumphs of NBG is that it is
finitely axiomatisable). We can, however, create a logical predicate
defining what it is to be a choice function. The version of the Axiom of
Choice I state is for global choice, and indeed this is the version used
by Kurt Godel. Therefore, this is the version I wish to work with.

This axiom is actually sort of instructive regarding models and the
difference between intuitive mathematics and formal mathematics.
Intuitively, sets exist, maybe classes exist too, and it's all good. We
don't wonder what the collection of all classes is. Formally, NBG is a
collection of axioms that only describe specific objects when interpreted
in a model. What's a model? It's an intuitive set (it can't be anything
else, since we are defining set theory!). That means that NBG's proper
classes are actually sets, but only in my mind; in the theory, they are
proper classes. So I can actually form, in my mind, the set of all choice
relations, which is not even a definable object in NBG but nonetheless
exists well enough to pick things out of for the purposes of satisfying an
existential axiom like the axiom of choice.

Okay, I lied; we are talking about internal versus external set theory.

--
Ryan Reich
ryan.reich@xxxxxxxxx
.

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