Shelah's Galois theory (memorandum)

"As an undergraduate, I found Galois theory beautiful, (more exactly what is in the book of Birkhoff-Maclane), and later I found Morley's theorem (with its proof) beautiful." -- Saharon Shelah, "The Future of Set Theory"

NOTICE

• DO NOT TRUST THE FOLLOWING

RESEARCH CONTEXT

• (added) Rather see Baldwin's slide on Abstract Elementary Classes
• (added) this paper might be helpful: An invitation to model-theoretic Galois theory
• (added) this might also be helpful: Une Théorie de Galois Imaginaire
• and my useless guessing follows

MOTIVATIONS

• Abelian variety is product of symmetric groups
• Galois theory states that solvable algebraic equations have their corresponding abelian groups
• Lie group is continuous, or infinite group
• Eigenvalue problem of (infinite dimensional) unitary operator should be "solvable" in certain meaning (gruppenpest)
• (added) Kakutani-Halmos: unitary oparator can be factorized as product of 4 symmetries, but not always as product of 3 symmetries
• What is Galois theory involving Lie groups?
• What is Galois theory involving groups of infinite degree?

HISTORY

• Sophus Lie and Friedrich Engel's Theorie Der Transformationsgruppen (1888) was motivated from Helmholtz' response to Riemann(1868)
• Cousin problems
• Whitehead conjecture
• Shelah's proof (V=L -> Whitehead group is free abelian group)

SPECULATIONS

• free group represents classical mechanics (zero curvature)
• non-free group represents "viscosity" of "elementary" mathematical objects
• regarding a cardinal as an infinite group which could be factorized into product of several symmetric infinite groups
• cofinality of singular cardinal might be degree of "normal subgroup" of this cardinal
• There's some arbitrariness in "factorization", or "cardinal division"
• (added) Such arbitrariness disappears when GCH holds (so under V=L too)

TODO

• clarify relations to Silver's works on singular cardinals (Silver was a pupil of Vaught, so his works also might have model theoretic implications)
• (added) "cardinal division" is inadequate. what is cofinality when GCH holds?
• (added) and how "ideal thoery of models" is?
• (added) clarify relations to entscheidungsproblem of transfinite functions (concerning complexity of elementary formula allowing recursively enumerable symbols of transfinite functions)