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Scores of expositions and technical analyses have been devoted to the materialization of the Zermelo-Russell antinomy in naive set theory as well as in Frege’s system of Grundgesetze der Arithmetik.1 The origins of the antinomy have been studied with great circumspection, and the impossibility of Frege’s attempted “Way Out” has been demonstrated, with essential participation of Polish logicians.2 Yet, there is no unanimity as to the ultimate cause of the deadlock up to now, and no full comprehension of Frege’s wavering between some hope for a conclusive repair of his system and the relinquishment of all hope for a secure foothold of our talk about classes, extensions of concepts, or—in his own terminology—courses-of-values. Reflection on the dilemma Frege found himself facing is certainly not out of place.

Delftse Foundations of Computation is a textbook for a one quarter introductory course in theoretical computer science. It includes topics from propositional and predicate logic, proof techniques, set theory and the theory of computation, along with practical applications to computer science. It has no prerequisites other than a general familiarity with computer programming.

Delftse Foundations of Computation is a textbook for a one quarter introductory course in theoretical computer science. It includes topics from propositional and predicate logic, proof techniques, set theory and the theory of computation, along with practical applications to computer science. It has no prerequisites other than a general familiarity with computer programming.

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, and making an estimate of future prospects. The symptom was the contradiction that had crept into naive set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege.

Based on the computer proof assistant Coq, the establishment of a formal system of naive set theory is realized, referring to Morse-Kelley set theory. On this basis, the formal descriptions of the axiom of choice and the product theorem of standard family sets are given, and the proof code for the equivalence between axiom of choice and the product theorem of the standard set family is completed and verified in Coq. All process fully embodies that machine proof of mathematical theorem based on Coq has the characteristics of readability and interactivity.

The chapter discusses the naive conception of set and criticizes attempts to rehabilitate it by modifying the logic of set theory. The focus is on the proposal that the Naive Comprehension Schema – which formally captures the thesis that every condition determines a set – is to be saved by adopting a paraconsistent logic. Three strategies for doing so are distinguished: the material strategy, the relevant strategy and the model-theoretic strategy. It is shown that these strategies lead to set theories that are either too weak or ad hoc or give up on the idea that sets are genuinely extensional entities.

To the memory of Prof. Grigori Mints, Stanford UniversityBorn: June 7, 1939, St. Petersburg, RussiaDied: May 29, 2014, Palo Alto, California

We are familiar with various set-theoretical paradoxes such as Cantor's paradox, Burali-Forti's paradox, Russell's paradox, Russell-Myhill paradox and Kaplan's paradox. In fact, there is another new possible set-theoretical paradox hiding itself in Wittgenstein’s Tractatus (Wittgenstein 1989). From the Tractatus’s Picture theory of language (hereafter LP) we can strictly infer the two contradictory propositions simultaneously: (a) the world and the language are equinumerous; (b) the world and the language are not equinumerous. I call this antinomy the world-language paradox. Based on a rigorous analysis of the Tractatus, with the help of the technical resources of Cantor’s naive set theory (Cantor in Mathematische Annalen, 46, 481–512, 1895, Mathematische Annalen, 49, 207–246, 1897) and Zermelo-Fraenkel set theory with the axiom of choice (hereafter ZFC) (Jech 2006: 3–15; Kunen 1992: xv–xvi; Bagaria 2008: 619–622), I outline the world-language paradox and assess the unique possible solution plan, i.e., the mathematical plan utilizing ‘infinity’. I conclude that Wittgenstein cannot solve the hidden set-theoretical paradox of the Tractatus successfully unless he gives up his finitism.

This paper extends the theory of situated inference from Mares (Relevant logic: a philosophical interpretation. Cambridge University Press, Cambrdge, 2004) to treat two weak relevant logics, B and DJ. These logics are interesting because they can be used as bases for consistent naive theories, such as naive set theory. The concepts of a situation and of information that are employed by the theory of situated inference are used to justify various aspects of these logics and to give an interpretation of the notion of set that is represented in the naive set theories that are based on them.

Transmathematics has the ambition to be a total mathematics. Many areas of the usual mathematics have been totalised in the transmathematics programme but the totalisations have all been carried out with the usual set theory, ZFC, Zermelo-Fraenkel set theory with the Axiom of Choice. This set theory is adequate but it is, itself, partial. Here we introduce a total set theory as a foundation for transmathematics. Surprisingly we adopt naive set theory. It is usually considered that the Russell Paradox demonstrates that naive set theory is incoherent because an apparently well-specified set, the Russell Set, cannot exist. We dissolve this paradox by showing that the specification of the Russell Set admits many unproblematical sets that do not contain themselves and, furthermore, unequivocally requires that the Russell Set does not contain itself because, were it to do so, that one element of the Russell Set would have contradictory membership. Having resolved the Russell Paradox, we go on to make the case that naive set theory is a paraconsistent logic. In order to demonstrate the sufficiency of naive set theory, as a basis for transmathematics, we introduce the transordinals. The von Neumann ordinals supply the usual ordinals, the simplest unordered set is identical to transreal nullity, and the Russell Set, excluding nullity, is the greatest ordinal, identical to transreal infinity. The generalisation of the transordinals to the whole of established transmathematics is already known. As naive set theory contains all other set theories, it provides a backwardly compatible foundation for the whole of mathematics.

In this paper, a single-antecedent/succedent sequent calculus NL for first-order Nelsonian paraconsistent quantum logic is investigated. The logic under consideration is regarded as a combination of both Nelson's paraconsistent four-valued logic and Dalla Chiara and Giuntini's paraconsistent quantum logic. The duality and cut-elimination theorems for NL are proved. Decidability, some constructive properties, some constructible falsity properties, and Craig interpolation property are shown for NL. An extend NL with some naive comprehension rules in the naive set theory is also investigated.

A logic for inconsistent mathematics must be strong enough to support reasoning in proofs, while weak enough to avoid paradoxes. We present a substructural logic intended to meet the needs of a working dialetheic mathematician—specifically, by adding a de Morgan negation to light linear logic, and extending the logic with a relevant conditional. The logic satisfies a deduction theorem. Soundness and completeness is established, showing in particular that contraction is invalidated. This opens the way for a robust naive set theory; we conclude by showing how the set theory provides a foundation for induction.

We discuss logical difficulties with the naive set theory based on the weak relevant logic DKQ. These are induced by the restrictive nature of the relevant conditional and its interaction with set theory. The paper concludes with some possible ways to mitigate these difficulties.

In mathematics, the creation and definition of new concepts is the first step in opening up a new field of research. Traditionally this step originated from intuition by a process of observation, analysis and abstraction. This article will show a general method by which most of the common notions of number theory, geometry, topology, etc., can be introduced in one and the same particular way. Therefore, we only need some of the tools of naive set theory: a set of terms to which we apply an equivalence relation. This equivalence relation induces a partition of the terms with which we can consequently associate new concepts. By using this method’s ‘machine’ in an abstract way on arbitrary sets of terms we can create new notions at will, as we will show in this article, for instance, for bidecimal numbers of different kinds. The fact that we reverse the usual procedure of intuition before abstraction, doesn’t mean that we only create esoteric objects without any meaning. On the contrary, their abstract nature precisely provides our imagination with many possibilities for several interpretations in models in which they become useful. So, for example, we can use our bidecimal numbers to define elementary transformations on a cylinder or on a pile of tori.

Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from a logic, but rather must be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments.

This paper probes into the foundations of conventional accounting measurements in order to construct a relatively simple axiom system on which a purely mathematical measurement system can be erected and thus provide a consistent basis for examining pertinent aspects of conventional accounting practices. […] We take conventional accounting measurement as given rather than, ab initio, seeking to prescribe what we think accounting measurement should be. […] Our attempt has been directed toward approximating conventional accounting by a relatively simple set of axioms and measurement rules, in the same manner that scientists in physics or chemistry have tried to develop a relatively simple set of concepts and theories in order to explain, in satisfactory degrees of approximation, complicated phenomena in this world.

Sets and sequences are the fundamental objects of study in discrete mathematics, and this chapter provides a review of naive set theory. Formulas for counting sequences, subsets, and permutations are developed, followed by an algorithmic method for counting when no formula applies (or is obvious). The final section looks at infinite sequences, particularly complexity functions of algorithms.

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe (Halmos, in: Naive set theory, 1960). In this paper, I am going to challenge this claim by taking seriously the idea that we can talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched.

Dialetheism is the metaphysical claim that there are true contradictions. And based on this view, Graham Priest and his collaborators have been suggesting solutions to a number of paradoxes. Those paradoxes include Russell’s paradox in naive set theory. For the purpose of dealing with this paradox, Priest is known to have argued against the presence of classical negation in the underlying logic of naive set theory. The aim of the present paper is to challenge this view by showing that there is a way to handle classical negation.

In a recent series of papers, I and others have advanced new logical approaches to familiar paradoxes. The key to these approaches is to accept full classical logic, and to accept the principles that cause paradox, while preventing trouble by allowing a certain sort of nontransitivity. Earlier papers have treated paradoxes of truth and vagueness. The present paper will begin to extend the approach to deal with the familiar paradoxes arising in naive set theory, pointing out some of the promises and pitfalls of such an approach.

One of the goals of this text is to get you proving mathematical statements in real analysis. Set theory provides a safe environment in which to learn about math statements, “if ... then,” “if and only if,” etc., and to learn the logic behind proofs. Since this is an introductory book on analysis, our treatment of sets is “very naive,” in the sense that we actually don’t define sets rigorously, only informally; we are mostly interested in how “they work,” not really what they are.

Prawitz observed that Russell’s paradox in naive set theory yields a derivation of absurdity whose reduction sequence loops. Building on this observation, and based on numerous examples, Tennant claimed that this looping feature, or more generally, the fact that derivations of absurdity do not normalize, is characteristic of the paradoxes. Striking results by Ekman show that looping reduction sequences are already obtained in minimal propositional logic, when certain reduction steps, which are prima facie plausible, are considered in addition to the standard ones. This shows that the notion of reduction is in need of clarification. Referring to the notion of identity of proofs in general proof theory, we argue that reduction steps should not merely remove redundancies, but must respect the identity of proofs. Consequentially, we propose to modify Tennant’s paradoxicality test by basing it on this refined notion of reduction.

This chapter contains a brief introduction to set theory which is essential for doing mathematics. There are two main axiomatic systems to introduce sets, viz. Zermelo–Fraenkel axiomatic system and the Godel–Bernays axiomatic system. We follow the Zermelo-Fraenkel axiomatic system together with the axiom of choice.

In order to solve a precise problem on trigonometric series, “Can a function have more than one representation by a trigonometric series” the great German mathematician Georg Cantor created set theory and laid the foundations of the theory of real numbers. This had a profound impact on mathematics. In this article, we narrate this fascinating story.

In Rogerson and Restall's (J Philos Log 36, 2006, p. 435), the "class of implication formulas known to trivialize (NC)" (NC abbreviates "naive comprehension") is recorded. The aim of this paper is to show how to invalidate any member in this class by using "weak relevant model structures". Weak relevant model structures verify deep relevant logics only.

We give some negative results on the expressiveness of naïve set theory (NS) in LP and in the four variants of minimally inconsistent LP defined in Crabbé (2011): ${\rm{L}}{{\rm{P}}_m},{\rm{L}}{{\rm{P}}_ = },{\rm{L}}{{\rm{P}}_ \subseteq }$, and ${\rm{L}}{{\rm{P}}_ \supseteq }$. We show that NS in LP cannot prove the existence of sets that behave like singleton sets, Cartesian pairs, or infinitely ascending linear orders. We show that NS is close to trivial in ${\rm{L}}{{\rm{P}}_m}$ and ${\rm{L}}{{\rm{P}}_ \subseteq }$, in the sense that its only minimally inconsistent model is a one-element model. We show that NS in ${\rm{L}}{{\rm{P}}_ = }$ and ${\rm{L}}{{\rm{P}}_ \supseteq }$ has the same limitations we give for NS in LP.

The aim of this paper is modern establishment of geometric theory of continuity, which is based on two, fourth group axioms - Archimedes ' and Cantor's axiom. Various consequences of Archimedes' and Cantor's axioms are proven such as Cantor's and Dedekind's theorem. The paper gives a special view on Hilbert 's axiomatic establishment of geometry which uses axiom of linear completeness instead of Cantor's axiom. Finally, the paper illustrates the use of the axioms of continuity and their consequences in proving some theorems.

The aim of this paper is to give an alternative definition of sets as follows: A domain is a if and only if it belongs to Set.

Logicians interested in naive theories of truth or set have proposed logical frameworks in which classical operational rules are retained but structural rules are restricted. One increasingly popular way to do this is by restricting transitivity of entailment. This paper discusses a series of logics in this tradition, in which the transitivity restrictions are effected by a determinacy constraint on assumptions occurring in both the major and minor premises of certain rules. Semantics and proof theory for 3-valued, continuum-valued and surreal-valued semantics are given and the proof theory for the systems outlined. The framework is robust in the sense that no conditional, defined or primitive, which sustains the contraction principles underlying Curry paradoxes can be expressed. Classical recapture is smoothly achievable in the system which however is expressively limited and not semantically closed. The conclusion considers the issue of how to extend the system to capture full naive set theory.

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. This leads to paradoxes analogous to those that afflict naive set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU.

The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches.

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naive set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u F = {x | F(x)} for a property F(.) could never be self-predicative in the sense of u F ∈ u F . But the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals – which can be seen as forming the “other bookend” to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato’s Theory of Forms as well as for the idea of a “concrete universal” in Hegel and similar ideas of paradigmatic exemplars in ordinary thought.

We introduce a subalgebra BC− of Bellantoni and Cook's safe-recursion function algebra BC. Functions of the subalgebra have safe arguments that are non-contractible (i.e non-duplicable). We propose a definition of safe and normal variables in light affine logic (LAL), and show that BC− is the largest subalgebra that is interpretable in LAL, relative to that definition. Though BC− itself is not PF complete, there are extensions of it (by additional schemes for defining functions with safe arguments) that are, and are still interpretable in LAL and so preserve PF closure. We focus on one such which is BC− augmented by a definition-by-cases construct and a restricted form of definition-by-recursion scheme over safe arguments. As a corollary we obtain a new proof of the PF completeness of LAL.

Chapter 1. Inconsistent Thinking, Fast and Slow Francesco Berto.- Chapter 2. Recursive functions for paraconsistent reasoners Zach Weber.- Chapter 3. Instantaneous Contradiction in Motion and Perception: Modeling the Phenomenal Present with a Dialetheic Logic of Time Corry Shores.- Chapter 4. Saving Proof from Paradox: Against the Inconsistency of Informal Mathematics Fenner Tanswell.-Chapter 5. Revenge for Berto's Law of Non-Contradiction Diego Tajer.- Chapter 6. On Coherence and Inconsistency Martin Pleitz.- Chapter 7. On the Preservation of Reliability Bryson Brown.- Chapter 8. Inconsistency Handling in the Sciences: Where and How do we Need Paraconsistency? Joke Meheus.- Chapter 9. Revision-Theoretic Truth and Degrees of Paradoxicality Cian Chartier.- Chapter 10. Inconsistent Scientific Theories: A Framework Otavio Bueno.- Chapter 11. Prospects for triviality Luis Estrada Gonzales.- Chapter 12. On the interpretation of classical mathematics in naive set theory Morgan Thomas.- Chapter 13. Doing Mathematics Paraconsistently. A manifesto. Maarten McKubre-Jordens.- Chapter 14. Why designate gluts? Andreas Kapsner.- Chapter 15. On the methodology of paraconsistent logic Heinrich Wansing and Sergei Odintsov.- Chapter 16. Dynamic proofs for networks of partial structures Holger Andres and Peter Verdee.

Around 1880, Georg Cantor, a German mathematician, invented naive set theory. A small fraction of this is sometimes taught to elementary school children. It was soon discovered that this naive set theory was inconsistent because it allowed unbounded set formation, such as the set of all sets. David Hilbert, the world's foremost mathematician from 1900 to 1930, defended Cantor's set theory but suggested a formal axiomatic approach to eliminate the inconsistencies.

I supply a simple key for ‘translating’ some semantical paradoxes into set-theoretical counterparts and vice-versa. This suggests a unified solution of these paradoxes, one which becomes particularly easy to see, because the ‘partners’ of some deep paradoxes are fairly trivial. One upshot of the investigation is that it is illicit, because circular, stipulation that gives rise to standard and ‘truth-teller’ paradoxes, and that there is a clear and natural restriction that prevents such circularity. In set theory, what we require are not new foundations for a consistent axiomatic system but only a minor restriction on the comprehension principle of naïve set theory that preserves our naïve intuitions.

Usually regarded as apparently true statements that seem to defy logic and lead to contradictions or as genuine phenomena that conflict with our theoretically based expectations, paradoxes comprise a related family of meanings that are often used rather loosely. In its looser sense, the term ‘paradox’ includes any statement or phenomenon that is more surprising than expected even when underlying theory is intuitive and has not been explicitly formulated. On the other hand, a famous example of paradox in its narrower or more technical sense is Bertrand Russell's (1872–1970) paradox or antinomy. The paradox indicated a fundamental problem with naive set theory because its assumption (plausible to many people) that it is permissible for sets to be or not to be members of themselves led to a contradiction. Thinking about the paradox stimulated modifications to set theory. Paradoxes are of great interest in science as well as in mathematical logic because they challenge those existing theories with which they conflict. Relative to predicted findings, paradoxical findings are therefore much more likely to advance fundamental theory. Three guidelines need to be followed in order to evaluate whether claims for paradoxes require modification of brain theories, and, if so, what these modifications might be. First, evidence for such phenomena and about their precise nature needs to be convincing. As, by definition, the phenomena conflict with our explicit or intuitive brain theories, this evidence needs to be much stronger than that for expected phenomena. Although this is reasonable, it is certainly frustrating for scientists who make highly original discoveries that conflict with current views. For example, the work of Czeisler et al. (1995) claiming that some blind people can reset their circadian rhythms using light for which they have no awareness, took 5 years and 20 rejections before it was eventually published. The …

In this paper, we look at how the notion of bijections can be used within the frame of Sergeyev’s numeral system. We give two definitions for counting the number of elements of a set and we explore the connections between these two definitions. We also show the difference between this new numeral system and the results of the traditional naive set theory.

The need for modifying axiomatic set theories was caused, in particular, by the development of category theory. The ZF and NBG axiomatic theories turned out to be unsuitable for defining the notion of a model of category theory. The point is that there are constructions such as the category of categories in naive category theory, while constructions like the set of sets are strongly restricted in the ZF and NBG axiomatic theories. Thus, it was required, on the one hand, to restrict constructions similar to the category of categories and, on the other hand, adapt axiomatic set theory in order to give a definition of a category which survives restricted construction similar to the category of categories. This task was accomplished by promptly inventing the axiom of universality (AU) asserting that each set is an element of a universal set closed under all NBG constructions. Unfortunately, in the theories ZF + AU and NBG + AU, there are toomany universal sets (as many as the number of all ordinals), whereas to solve the problem stated above, a countable collection of universal sets would suffice. For this reason, in 2005, the first-named author introduced local-minimal set theory, which preserves the axiom AU of universality and has an at most countable collection of universal sets. This was achieved at the expense of rejecting the global replacement axiom and using the local replacement axiom for each universal class instead. Local-minimal set theory has 14 axioms and one axiom scheme (of comprehension). It is shown that this axiom scheme can be replaced by finitely many axioms that are special cases of the comprehension scheme. The proof follows Bernays’ scheme with significant modifications required by the presence of the restricted predicativity condition on the formula in the comprehension axiom scheme.

International audience; Our aim is to quantify how complex is a Cantor set as we approximate it better and better. We formalize this by asking what is the shortest program, running on a universal Turing machine, which produces this set at the precision ε in the sense of Hausdorff distance. This is the Kolmogorov complexity of the approximated Cantor set, that we call the "ε-distortion complexity". How does this quantity behave as ε tends to 0? And, moreover, how does this behaviour relates to other characteristics of the Cantor set? This is the subject of the present work: we estimate this quantity for several types of Cantor sets on the line generated by iterated function systems (IFS's) and exhibit very different behaviours. For instance, the ε-distortion complexity of a C^k Cantor set is proven to behave like ε^{−D/k}, where D is its box counting dimension.