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In his ASL Godel lecture (Las Vegas, Nevada, 2002), Harvey Friedman asked the following question:

Are there fundamental principles of a general philosophical nature which can be used to give consistency proofs of set theory, including the so called large cardinal axioms?

Recently, he is able to isolate the following two fundamental philosophical principles and use them to prove the interpretability of various common sense thinking and set theory (with large cardinals).

(i) Plenitude Principle (PP): Anything that can happen will.

(ii) Indiscernibility Principle (IP): Any two horizons are indiscernible to observers on the basis of their extent.

(See his Concept Calculus article, e.g. link text)

My main questions are: How confident are we that PP and IP are “true” ? More specifically, is it possible to “prove” or justify PP and IP rigorously? If yes, how? If not, why not?

In my view, the ultimate justification of PP and IP would be to construct a (meta) system S based on PP and IP, and then prove its consistency and completeness. In the light of Godel’s incompleteness theorem, I’m not sure that this can be done. But perhaps S is not recursively axiomatizable, and so Godel’s incompleteness theorem would not apply to S.

My secondary question is: are there any logician (beside Friedman) who are working on this kind of research?

Update: July 2011

Here is a rephrasing of the question by Timothy Chow that makes it closer to mathematical logic:

Is there some precise mathematical statement, that has the flavor of IP or PP, which proves the consistency of all (or most) set-theoretic axioms that are generally accepted today (e.g., large cardinal axioms)?

Update: The question has now been open. It is now time for people who can relate to the problem to answer it.

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This is not a mathematical question. It's also not a meaningful question, in my opinion, but at any rate the former reason suffices for it to be inappropriate for MO. –  Zev Chonoles Nov 12 '10 at 2:53
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I don't agree with Zev - I am not sure if the question is meaningful, but I think it should be judged by people who have more experience than me with the practice and current concerns of set theory or logic. Casting a vote against closing. –  Yemon Choi Nov 12 '10 at 3:11
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I would ordinarily not be so bold as to make a strong claim about a question in an area outside my interests, but this kind of attempt to directly relate philosophical concepts with mathematical ones reminds me of en.wikipedia.org/wiki/Alain_Badiou#Mathematics_as_ontology. Of course, I will gladly defer to a set theorist's or logician's judgment about this question. –  Zev Chonoles Nov 12 '10 at 3:37
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There's a thread on meta about reasons for/against closing this question: tea.mathoverflow.net/discussion/764/… –  Anton Geraschenko Nov 12 '10 at 4:23
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I think there may be a math problem in here, but the posting fails to say what it is, unless the linked writings explain it. –  Michael Hardy Nov 12 '10 at 16:57

1 Answer 1

In a more recent paper Friedman defines a mathematically precise system MBT (much better than) and proves it and ZF have mutual interpretability. This establishes that if either is consistent they both are. These axioms have some of the flavor of IP and PP, but of course these axioms are not implied by IP and PP.

At the end of the paper Friedman claims a to be published result. STAR is defined as:

There exists a star. I.e., something which is better than something, and much better than everything it is better than.

We have shown that MBT + STAR can be interpreted in some large cardinals compatible with V = L, and some large cardinals compatible with V = L are interpretable in MBT + STAR."

For PP and IP to be true, in a sense that can prove mathematics, they need to be stated precisely like the axioms in MBT. That formulation is much more complex than PP and IP as it must be to interpret ZF.

It is important to keep in mind that consistency does not imply truth. The statement that a formal system is consistent is equivalent to a statement of the form $\forall_{n\in\omega} r(n)$ where $r$ is a recursive relationship. This is equivalent to the halting problem for a particular Turing machine.

The following quote from Friedman is, I suspect, a big part of his and others interest in this work:

STARTLING OBSERVATION. Any two natural theories S,T, known to interpret PA, are known (with small numbers of exceptions) to have: S is interpretable in T or T is interpretable in S. The exceptions are believed to also have comparability.

It is an interesting and even startling observation, but it is worth keeping in mind that that rigorous theories are, among other things, recursive processes for enumerating theorems. To say that one theory is interpretable in another is to say a subset of one processes outputs are, in a specific well defined sense, isomorphic to the outputs of the process defined from the theory being interpreted. Whatever other significance it may have, this is a statement about unbounded recursive processes.

My personal view (see what is Mathematics About?) is that the only mathematics that can be interpreted as a properties of recursive processes is objectively true or false. This is based on the old idea that infinite is a potential that can never be realized. In this view Cantor's proof that the reals are not countable is an incompleteness theorem. The cardinal hierarchy is a hierarchy of the ways the real numbers provably definable in a formal system can always be expanded. Because of the Lowheheim Skolem theorem, we know such an interpretation exists. Interpretations that assume the absolutely uncountable are inevitably ambiguous at least as far as they can be expressed formally in the always finite universe that we seem to inhabit.

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