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J. Hagendorf asked if every
order type ϕ having the following two properties of additively indecomposable
ordinals was the order type of an ordinal. Call ϕ Hagendorf if (i) it is strictly
indecomposable to the right, i.e., if ϕ = ψ + 𝜃, then ϕ can be embedded in 𝜃 but not
in ψ, and (ii) every strictly smaller type can be embedded in an initial segment of ϕ,
i.e., if χ can be embedded in ϕ but not vice versa, then ϕ = ψ + 𝜃 where 𝜃≠0 and χ
can be embedded in ψ. Recall that scattered order types are those which do not
embed the order type of the rationals.
The paper provides a partial answer to Hagendorf’s question: Every scattered
Hagendorf type is the order type of an indecomposable ordinal.
Other subclasses of order types for which this question seems particularly
interesting are sub-types of the order type of the real numbers, and the class of
countable unions of scattered types.
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