We study the interpretation of Grzegorczyk’s Theory of Concatenation
TC in structures of decorated linear order types satisfying Grzegorczyk’s
axioms. We show that TC is incomplete for this interpretation. What
is more, the first order theory validated by this interpretation interprets
arithmetical truth. We also show that every extension of TC has a model
that is not isomorphic to a structure of decorated order types.
We provide a positive result, to wit a construction that builds structures
of decorated order types from models of a suitable concatenation
theory. This construction has the property that if there is a representation
of a certain kind, then the construction provides a representation of
that kind
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