User lianna - MathOverflow

Philosophical Consistency Proof for Set Theory

2010-11-12T02:21:01Z

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.

Plenitude Principle and Mathematics

2010-11-12T11:50:00Z

I think closing my previous question link text on the basis of its not being mathematics would be a mistake. At least there are two famous theorems of contemporary mathematics echo the principle of plenitude (PP), namely The Infinite Monkey Theorem and Kolmogorov's zero-one law (See e.g. link text).

In his most recent posting on FOM, Harvey Friedman used PP to formulate the system OEU and proved the mutual interpretability of ZF and OEU. That means anything that you can prove in ZF, you can prove in OEU as well. Therefore if mathematics can be interpreted in ZF, then mathematics can be interpreted in OEU as well. See his complete posting on FOM 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?

Comment by Lianna

2010-11-24T11:19:46Z

@Ed Dean: Yes of course Godel's incompleteness applies to recursively enumerable theory. That's clear, but that's not my question. Stefan comment seems to imply that Godel's incompleteness applies to non-recursively theory as well, he says: "Maybe you could get away with a recursively enumerable set of axioms, but that also cannot give you a complete theory". I don't understand this statement because Godel's incompleteness would not apply to NON-recursively enumerable theory.

Comment by Lianna

2010-11-24T02:42:32Z

(Maybe you could get away with a recursively enumerable set of axioms, but that also cannot give you a complete theory.) Stefan, can you explain why non-recursively axiomatizable theory cannot be complete? In my understanding, Godel's incompleteness theorem would not apply to non-recursively axiomatizable theory.