Abstract
Hirschfeld and Wheeler proved in 1975 that ∑1 ultrapowers (= “simple models”) are rigid; i.e., they admit no non-trivial automorphisms. We later noted, essentially mimicking their technique, that the same is true of Δ1 ultrapowers (= “Nerode semirings”), a class of models of Π2 Arithmetic that overlaps, but is mutually non-inclusive with, the class of Σ1 ultrapowers. Hirschfeld and Wheeler left as open the question whether some Σ1 ultrapowers might admit proper isomorphic self-injections. We do not answer that question; but we do answer the corresponding question, in the negative, for the Δ1 case.
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References
Hirschfeld J. (1975). Models of arithmetic and recursive functions. Israel J. Math. 20(2): 111–126
Hirschfeld J. and Wheeler W. (1975). Forcing, arithmetic, division rings. Lecture Notes in Mathematics 454. Springer, Berlin
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