Abstract
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.
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References
GoguenJ., ThatcherJ., WagnerE., and WrightJ.:J. Assoc. Comput. Mach. 24 (1977), 68.
HofstaedterD.:Gödel, Escher, Bach, Basic Books, New York, 1979.
KauffmanL. and VarelaF.:J. Biol. Soc. Struct. 3 (1980), 171.
LawvereF. W.: Diagonal arguments in cartesian closed categories, inCategory Theory. Homology Theory and their Applications II, Lecture Notes in Mathematics, Vol. 92, Springer-Verlag, New York, 1969, pp. 134–145.
ScottD.: in,Semantics of Algorithmic Languages, Lecture Notes in Mathematics188, Springer-Verlag, New York, 1971.
ScottD.: in,Topoi, Algebraic Geometry and Logic, Lecture Notes in Mathematics,274, Springer-Verlag, New York, 1972.
ScottD.: in,Proc. 4th. Int. Congress Logic, North Holland, Amsterdam, 1973.
VarelaF.:Principles of Biological Autonomy, North Holland, New York, 1979.
MandelbrotB.:Fractals: Form, Chance, Dimension, Freeman, San Francisco, 1977.
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Soto-Andrade, J., Varela, F.J. Self-reference and fixed points: A discussion and an extension of Lawvere's Theorem. Acta Appl Math 2, 1–19 (1984). https://doi.org/10.1007/BF01405490
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DOI: https://doi.org/10.1007/BF01405490