“List all computable functions that it can prove are well defined, and diagonalize over them”

What means “computable function” ? Do you mean that it is about making a list of all algorithms written in a given (turing-machine idealized) computer language that produce a series of numbers until they eventually stop (run forever and no more give a next number), and then extract the sublist of all such algorithms that a given theory (like ZF with Reinhardt cardinal) proves will never stop issuing more numbers ?
OK, let’s do this way. Diagonalize over this and take the value of this function at a trillion.
If this is ordered by length of algorithms, then it would be a non-algorithmic definition, as it depends on provability questions that are not always decidable.
A problem with this definition of a very large natural number, is that the value of the number it defines depends on the universe in which you interpret it.
Indeed, let’s see what may happen if you interpret it in a universe with a non-standard set of integers. Then it may contain a proof of non-standard length that some algorithm before your trillionth, will always issue numbers; this claim may be true as well but not admit a proof of standard length, so that this creates no contradiction. Once accepted from this proof of non-standard length that the given algorithm defines a function, it is included in the diagonalization.
Then your trillionth value interpreted in the non-standard universe may be higher than the one satisfying the same definition interpreted in a standard universe.
But you probably meant instead you ordered the algorithms by length of proof, in which case the final result will be computably defined as well (but just unprovably inside the same theory).

Now, back to the initial question of what means “computable function”. If one tried to do any less constructive and thus more powerful computation than the one of an algorithm, then the diagonalization makes no sense, as having a proof that a function is well-defined does not give its value, which will depend on the universe in which this definition is interpreted.

There is one problem: it is not clear that this function is well defined, and there is no known way of proving that it is. This is inevitable: any function that one can see is well defined must be quite small.

Quite small compared to these obscenely large things we can’t prove well-defined, that is!

Is there any interesting way to formulate a quest for extremely rapidly-growing functions — and thus extremely large numbers — that we can still prove are well-defined, using constructive methods?

When Peano Arithmetic is considered with the von Neumann construction, do we enter the arena of game theory mathematics?

Construction of the natural numbers in set theory
in Peano axioms
http://en.wikipedia.org/wiki/Peano_arithmetic

These equations may be treated as games:
x = y or x + (-y) = 0, a zero-sum game;
x > y or x + (-y) > 0, a non-zero-sum game.

welcome as an actor in this crazy exciting stage which is the blogosphere. I was only able to understand half of your post, but I found it intriguing, and I hope you will write also for the rest of us, with the insight you and only few others have.

Best regards,
T.

In fact, ε0 measures the strength of Peano arithmetic in a somewhat stronger sense : namely, in Peano arithmetic well-orderness of this ordinal implies
consistency. (The way it is done as follows: you can measure complexity of proofs in Peano arithmetic by ordinals less than ε0, and than “streamline” proofs by “a cut-elimination” procedure reducing the complexity and formalisable in PA. It is obvious that a “streamlined” proof cannot be a proof of a contradiction; you use well-orderness of ε0 to prove, uniformly, that every proof can be “streamlined”.) I would assume that this holds for some other theories as well.

In fact, one can define a very simple and explicit inductive process, which given a natural number, always converges to 0; but the number of iterations required bounds any function which PA proves to be totally defined. (A reference for this is a textbook of Takeuti on proof theory, _2nd_ edition.)

[crossposted from 7 May 2007 Good Math Bad Math
http://scienceblogs.com/goodmath/2007/05/basics_sets_and_classes.php

Last week I went to a special memorial lecture at
Caltech by William Hugh Woodin, [on] recent surprising
results in axiomatization of transfinite numbers. He
had been a professor at Caltech, where my degree in
Math included advanced infinitary set theory. Since
computers did not help at all in that field, I shifted
to CS and Mathematical Biology and other things.

As Wikipedia reports: “William Hugh Woodin (b. 1955,
Tucson, Arizona) is a set theorist at University of
California, Berkeley. He has made many notable
contributions to the theory of inner models and
determinacy. His recent work on Ω-logic suggests
an argument that the continuum hypothesis is false.”

“He earned his Ph.D. from UC Berkeley in 1984 under
Robert M. Solovay. His dissertation title was
Discontinuous Homomorphisms of C(Omega) and Set
Theory.”

“He served as chair of the UC Berkeley mathematics
department for the 2002-2003 academic year.”

“He is the great-grandson of William Hartman Woodin,
former Secretary of the Treasury.”

I asked him about unproven hunches by the recently
deceased great Mathematician Paul Cohen, who proved
the independence of the Cintinuum Hypothesis, and
who’d gone to my high school (Stuyvesant, New York).

Hugh Woodin was pleased that I brought this up, and
explained that he’d had many conversations with Cohen
about this, including in this past year, Cohen’s last.

The new axioms used weird combinatorics and infinite
games, and seemed tantalizingly close to making large
ordinals and cardinals as unambiguous and the
integers.

Without this lecture, I’d have been able to make
little sense of Wikipedi’a short article on “Woodin
cardinals.”

“… Woodin cardinals are important in descriptive set
theory. Existence of infinitely many Woodin cardinals
implies projective determinacy, which in turn implies
that every projective set is measurable, has the Baire
property (differs from an open set by a meager set,
that is, a set which is a countable union of nowhere
dense sets), and the perfect set property (is either
countable or contains a perfect subset).”

“… The consistency of the existence of Woodin
cardinals can be proved using determinacy hypotheses.
Working in ZF+AD+DC one can prove that Θ0 is
Woodin in the class of hereditarily ordinal-definable
sets. Θ0 is the first ordinal onto which the
continuum cannot be surjected…”

ZF+AD+DC defined as follows (again, thanks 3 times to
Wikipedia):

ZF = Zermelo-Fraenkel set theory (without the Axiom of
Choice = AC)

AD = the axiom of determinacy in set theory. “It was
introduced by Polish mathematicians: Mycielski and
Steinhaus. It states the following”:

“Consider infinite two-person games with perfect
information. Then, every game of length ω where
both players choose integers is determined, i.e., one
of the two players has a winning strategy.”

“The axiom of determinacy is inconsistent with the
axiom of choice (AC); indeed, it has been shown that
it implies that all sets of reals are Lebesgue
measurable and have the property of Baire.”

“AD implies the consistency of ZF. Hence it is not
possible to prove in ZF that ZF is consistent with
AD.”

DC = the axiom of dependent choices, “a weak form of
the axiom of choice (AC) which is still sufficient to
develop most of real analysis. Unlike full AC, DC is
insufficient to prove (given ZF) that there is a
nonmeasurable set of reals, or that there is a set of
reals without the property of Baire or without the
perfect set property.”

“DC is (over the theory ZF) equivalent to the
statement that every (nonempty) pruned tree has a
branch. It is also equivalent[1] to the Baire category
theorem for complete metric spaces.”

For a while, I was fascinated again by transfinite
visions. But it’s too hard for me. I thanked him, and
returned to the finite cosmos.