Avoiding the Paradoxes

www.ramoncasares.com 20250930 Scott 1 Avoiding the Paradoxes Ramo´n Casares orcid: 0000-0003-4973-3128 There is not any effective way of avoiding the paradoxes. Therefore, in particular, every paradox-free axiomatization of the concept set is incomplete. Keywords: set theory, paradoxes, incompleteness. §1 Introduction ¶1·In “Axiomatizing Set Theory”, page 208, Scott (1974) wrote: “The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the theory of types.” I cannot agree with Scott on this. Contrariwise, I will argue that there is not any satisfactory way of avoiding the paradoxes, not even an effective one. §2 Paradoxes ¶1·Scott (1974), page 207, cited Russell’s paradox, which is based on Russell’s set R: the set of all sets that are not members of themselves. Set Ris paradoxical because we cannot decide whether Rbelongs to Ror not. ¶2·Russell’s paradox is the liar paradox for sets, where the liar paradox is: ‘this sentence is not true’. Now we cannot decide whether that sentence is true or not. All these liar kind of paradoxes close a simple yes-no infinite loop: if yes then no, and if no then yes, and then no, and then yes, and so on forever. As it is easy for us to spot these simple infinite loops, they are not regarded as disasters, but more frequently paradoxes are just funny instances of abnormal objects. For example, we can say that Russell’s set Ris paradoxical and that paradoxical sets are not proper sets, or even that they are not sets at all, and then we could call them classes, or collections, or whatever, in order to keep our peace of mind. This is easy enough, so why should we avoid the paradoxes? This is doi: 10.6084/m9.figshare.28457378.v3, version 20250930. c 2025 Ramo´n Casares; licensed as cc-by. Any comments on it to [email protected] are welcome. www.ramoncasares.com 20250930 Scott 2 §3 Two ways ¶1·There are two canonical ways of dealing with the paradoxes: the avoiding way and the inclusive way. ¶2·The avoiding way uses a language where paradoxes cannot be defined. This is, for example, the theory of types way mentioned above by Scott. This way all objects are proper objects, so we will not be troubled by any paradox. So far, so good. ¶3·The inclusive way uses an expressive language, one in which all objects can be defined, and a procedure to filter out the paradoxes. This is the case of English, for example, where we can define Russell’s set R, and where we can show that Russell’s set Ris paradoxical. Then we can use an expressive language, and we can keep the troubling paradoxes apart. So far, so good. §4 Languages ¶1·Since the only computations that result undecided are those that do not halt, then, by definition, a decidable language is one in which every computation halts (after a finite number of steps). There are not infinite loops in the decidable languages. Therefore, there are not paradoxes in the decidable languages. ¶2·Under Church’s thesis, which we assume, the most expressive languages are the Turing complete languages, see Casares (H), also known as complete languages. By definition, acomplete language is the language of a universal Turing machine, see Casares (G). Any possible computation can be expressed in a complete language, see Casares (N). Therefore, any filter expressible in any language can be expressed in a complete language. §5 Argument ¶1·Simplifying a bit, the avoiding way of dealing with the paradoxes is asociated with the decidable languages, and the inclusive way with the complete languages. Though in principle both ways can deal with the paradoxes, none of them is effective. The reason is that, as a consequence of the halting theorem, there is no program that can identify just every paradox, since no program can decide for every computation whether it will halt or not, see Casares (N). And this implies both that no decidable language can express just everything that is not a paradox, and that no complete language, and then no language, can express a predicate to filter out just every paradox. ¶2·All this means that the avoiding way will avoid some proper objects, and that the inclusive way will include some paradoxes. Summarizing, either by lack or by excess, there is not any effective way of avoiding the paradoxes. §6 Conclusion ¶1·The previous summary does not rely on the canonical ways, because the halting theorem applies to every computable way of dealing with the paradoxes, where computable equals effective under Church’s thesis. In any case, the conclusion is that, if we can find a liar kind of paradoxical instance of a concept in an expressive language, then the concept is undecidable and consequently any avoiding way of dealing with that concept will be also avoiding some proper instances of it. Therefore, in particular, every paradox-free axiomatization of the concept set is incomplete. www.ramoncasares.com 20250930 Scott 3 References Casares (G): Ramo´n Casares, “Go¨del Incompleteness and Turing Completeness”. doi: 10.6084/m9.figshare.25434994. Casares (H): Ramo´n Casares, “A Complete Hierarchy of Languages”. doi: 10.6084/m9.figshare.6126917. Casares (N): Ramo´n Casares, “On Unprecise Numbers”. doi: 10.6084/m9.figshare.29066705. Scott (1974): Dana Scott, “Axiomatizing Set Theory”; in Proceedings of Symposia in Pure Mathematics, Volume 13, Part II, pages 207–214. 1974. doi: 10.1090/pspum/013.2/0392570.