Difference between revisions of "Automorphisms of countable recursively saturated models"
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| − | Let G be the automorphism group of a countable recursively saturated model of PA. For every nontrivial f∈G there is a countable recursively saturated elementary end extension N such that f has no extension to an automorphism of N. It is open whether there is an f∈G such that for all countable recursively saturated elementary end extensions N f does not extend to an automorphism of N? | + | Let G be the automorphism group of a countable recursively saturated model of PA. For every nontrivial f∈G there is a countable recursively saturated elementary end extension N such that f has no extension to an automorphism of N. It is open whether there is an f∈G such that for all countable recursively saturated elementary end extensions N, f does not extend to an automorphism of N? |
If S is an inductive partial satisfaction class of M, and f∈Aut(M,S), then there is a countable recursively saturated elementary end extension N such that f extends to an automorphism of N. If M is arithmetically saturated, then there are f∈G such that for every inductive partial satisfaction class S of M, f∉Aut(M,S). Problem: what if M is not arithmetically saturated? | If S is an inductive partial satisfaction class of M, and f∈Aut(M,S), then there is a countable recursively saturated elementary end extension N such that f extends to an automorphism of N. If M is arithmetically saturated, then there are f∈G such that for every inductive partial satisfaction class S of M, f∉Aut(M,S). Problem: what if M is not arithmetically saturated? | ||
Revision as of 09:48, 18 January 2013
Extending automorphisms
Let G be the automorphism group of a countable recursively saturated model of PA. For every nontrivial f∈G there is a countable recursively saturated elementary end extension N such that f has no extension to an automorphism of N. It is open whether there is an f∈G such that for all countable recursively saturated elementary end extensions N, f does not extend to an automorphism of N?
If S is an inductive partial satisfaction class of M, and f∈Aut(M,S), then there is a countable recursively saturated elementary end extension N such that f extends to an automorphism of N. If M is arithmetically saturated, then there are f∈G such that for every inductive partial satisfaction class S of M, f∉Aut(M,S). Problem: what if M is not arithmetically saturated?
