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