Conclusion

Self-reference is an entanglement of logical types that occurs when these are folded back upon themselves and fuse, like when an indication comes to point at itself. Self-referential situations produce paradoxical results easily, as in Epimenides’ dictum of the lying Cretans or as in Russell's barber paradox. An indication that refers to itself has been introduced in our definition of a one-sided boundary and this led us to the idea that paradox can be forestalled if boundaries are allowed to move outwardly. We argued that the expansion speed of these boundaries is necessarily finite and that the many encounters and interactions between expanding forms cannot be computed beforehand. Their course of development is predictable only for relatively short intervals. The duration of these intervals cannot be predicted either. The expanding forms are the outcomes of geometrical constructions rather than of computations, which gives the expansion processes an appearance of sensitive movements.