Harvey Friedman: The expanding mind

Question

In reference 1, Friedman writes:

I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel.

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B. 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?

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B3. THE EXPANDING MIND/2.

Here we obtain substantial strength with only two minds, M and the more powerful M*. M has its domain of objects d(M), and M* has its domain of objects d(M*), where every object of M is an object of M* but not vice versa. Thus d(M*) is richer than d(M).

One measure of the strength of a mind is the unary/binary relations on its domain that the mind M can define. Those relations are given by mental constructions of M that output the truth value of the relation at arguments from d(M). The same relation may be given by different mental constructions, as relations are extensional and constructions are intensional.

I assume that every mental construction of M is a mental construction of M*. Its range of applicability under M is d(M) and its range of applicability under M* is d(M*).

I use a very strong form of comprehension for the unary/ binary relations defined by M. M can use not only M* but any individual unary/binary relations defined by M*, for the purposes of defining unary/ binary relations.

I assume that M* fully dominates M in the sense that there is a binary relation defined by M* whose cross sections are the unary relations defined by M.

I also assume that M and M* agree on the truth values of all appropriate statements with parameters from the objects of M and the mental constructions of M.

The interpretation power of this philosophical story lies between a proper class of Woodin cardinals and an elementary embedding from a rank +1 into a higher rank +1.

This story can be extended in an appropriate way to an infinite sequence of minds M,M*,M**,... . This leads to a philosophical story of interpretation power between n-huge cardinals and a rank into itself.

Where can I find a formal exposition of this theory and more details about it, including proofs of the bolded statements?

References:

1. Harvey M. Friedman. Issues in the Foundations of Mathematics. 2002/06/02.

Answer 1

Friedman has made public on his website a 2016 draft titled "Expanding Mind Theory". On page 4 there is a definition of a theory $\mathrm{EM}$ in first-order logic which formalizes the theory of two minds.

Neither of the bold statements have proofs appearing in this document. There are two theorems working toward calibrating the logical strength of $\mathrm{EM}$ as in the first bold statement: theorem 2.2, that $\mathrm{ZF}$+"there exists a $1$-extendible cardinal" proves consistency of $\mathrm{EM}$, and theorem 3.2.19, that $\mathrm{EM}$ interprets $\mathrm{ZFC}$. The final section hints toward the existence of strengthened versions of this theory as in second bold statement, except with still two minds rather than infinitely many.