Self-reference and fixed points: A discussion and an extension of Lawvere's Theorem

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We consider an extension of Lawvere's Theorem showing that all classical results on limitations (i.e. Cantor, Russel, Godel, etc.) stem from the same underlying connection between self-referentiality and fixed points. We first prove an even stronger version of this result. Secondly, we investigate the Theorem's converse, and we are led to the conjecture that any structure with the fixed point property is a retract of a higher reflexive domain, from which this property is inherited. This is proved here for the category of chain complete posets with continuous morphisms. The relevance of these results for computer science and biology is briefly considered.