[FOM] query about Peter Koellner's thesis "The Search for New Axioms"
Rupert McCallum
Rupert.McCallum at acu.edu.au
Fri Mar 12 17:34:54 EST 2010
In Peter Koellner's thesis "The Search for New Axioms", he argues for an axiom EP, which says that if alpha is an ordinal, then there exist ordinals kappa and lambda such that alpha<kappa<lambda and a non-trivial elementary embedding j:L(V_kappa)->L(V_lambda). He argues for this by arguing that there ought to be a sequence S of ordinals cofinal in On which form a class of order indiscernibles and such that {V_alpha:alpha in S} is a class of "candidates for V" which satisfy "all the closure properties implied by the concept of set".
It seems to me that this can be formalised in BGC+"On is Ramsey". One considers the structure (V,epsilon) augmented by Skolem functions and obtains a proper class of ordinals which are indiscernibles for this structure. One then has an elementary embedding j:M->M where M is the Skolem hull of this class of ordinals, and M in turn is elementary equivalent to V. The ordinals in the class are all extremely indescribable and totally ineffable, and so forth. Hence V_alpha where alpha is an ordinal in the class can be thought of as a "candidate for V". It seems to me that we can then prove EP along the lines suggested by Koellner in the thesis.
However, I am worried because Koellner also claims that one can prove along similar lines that the Dodd-Jensen core model is non-rigid because it is built up from below in a "definable fashion". This would imply the existence of L[U] and so this cannot be formalised in BGC+"On is Ramsey" unless the existence of Ramsey cardinal is inconsistent. So either there is some aspect of the reasoning which cannot be formalised in BGC+"On is Ramsey", or else there is an error in the thesis. I do not know which because I am not very familiar with the theory of the core model. I was wondering if anyone could help me out here.
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