The target of this chapter is to give some rudiments of(paraconsistent) real analysis, a dissection of the linear continuumas that which has no gaps. Axioms for the reals as a totally orderedcomplete field are given and developed, with philosophicalconsiderations about the nature of points. Focus is on the topologyof the real line, establishing the general principle at stake: thatif a change occurs, it must occur somewhere. This is confirmed atthe intermediate value theorem. Along the way, the (continuous)sorites paradox is “recaptured,” the existence ofinfinitesimal quantities is floated, and a theorem is proved about“splitting” geometric points.

This chapter transitions from the foundational topics of the previoustwo chapters, to more “working” mathematics. Thischapter builds up intuitions about some simple algebraic concepts– groups, rings, and fields. Some diagrams of“inconsistent vectors” are played with. The discussionis organized around how to deal with an open problem in inconsistentmathematics from Dunn and Mortensen, of how to allow some nontrivialinconsistency in fields. The idea of using relative unit elements ingroup cancellation – more than one “zero”number – is introduced as a possible solution, and its basicrules are developed.

This chapter looks at the methodology for dealing with paradoxes vianonclassical logic. Some standard approaches by Field, Beall, andPriest (among others) are considered and found unsatisfactory forrelying on classical logic in the “metatheory.” Theidea of paraconsistency as a universal logic is suggested instead,and the main challenge for this approach is presented: the“Feferman objection” that nonclassical logics cannotdo mathematics. This leads to a discussion of “classicalrecapture” and mathematical revisionism. The plan for using aparaconsistent “metalanguage” is outlined, the“just true” challenge is dismissed, and the goals forthe approach are laid out – namely, to develop enoughmathematics to establish the paradoxes from Chapter 1.

This concluding chapter draws together some threads of the book, byfacing some objections about negation and more generally about theconceptions of truth, falsity, validity, invalidity, and theirinterrelations. The main question – why are there paradoxes?– is asked three last times.

This chapter introduces paradoxes from three areas. First, the famousantinomies from Frege–Cantor set theory are presented asconsequences of the naive set concept; the standard solution,Zermelo–Fraenkel set theory and the idea of iterative sets,is discussed and found to be problematic. Second, the soritesparadox for vague predicates/properties is presented, and somestandard solutions are discussed, putting the focus on“cutoff points.” A paraconsistent “glut”approach is recommended as the best way to accept the existence ofcutoff points, as inconsistent and not unique. Third, a puzzle aboutboundaries in space is presented, standard solutions discussed andfound wanting, and a paraconsistent response outlined. It issuggested that all three paradoxes are interrelated by the conceptof “revenge” and should be met with dialetheism.

This chapter considers two possible explanations for the paradoxes.One is Lawvere’s diagonal theorem from category theory. Theother is the inclosure schema, proposed by Priest as the structureof many paradoxes and a step toward a uniform solution to theparadoxes. Inclosure suggests that paradoxes arise at the limits ofthought because the limits can be surpassed, and also not. Theconsequences of accepting Priest’s proposal are explored, andit is found that, from a thoroughly dialetheic perspective, (i) somelimit phenomena cannot be contradictory, on pain of absurdity, and(ii) true contradictions are better thought of as local, not“limit,” phenomena. Dialetheism leads back from theedge of thought, to the inconsistent in the everyday.

This chapter introduces the liar paradox from naïve truththeory, and the central dilemma for solving it: any theory of truthwill be incomplete or inconsistent. Various incompleteness optionsfrom classical and paraconsistent logics are considered. Dialetheicparaconsistency – the idea that there are true contradictions– is introduced as an alternative inconsistent option, andits main logical and philosophical features are presented. Aproblematic fixed point theorem is introduced as a possible key toexplaining the paradoxes.

This chapter follows the axiomatic development of elementaryarithmetic. Starting from Peano’s postulates, somefundamental properties of natural numbers are derived, concerningaddition, multiplication, and order. If further assumptions aboutmathematical induction are made, then famous results about primenumbers can be established, including the fundamental theorem ofarithmetic (unique prime factorization) and Euclid’s theoremabout the infinitude of primes. Some simple facts about even and oddnumbers lead to the original limitive diagonal theorem: that thesquare root of 2 is irrational. Features of working withoutcontraction are highlighted.

In this chapter, the basic theory of sets is developed axiomaticallyin a paraconsistent logic. The two main goals are (1) to establish atoolkit for elementary mathematics, and (2) to prove the mainantinomies of naive set theory. The two goals come together inproving the Burali-Forti paradox for the theory of ordinals. Alongthe way, results are proved about the universal set, various formsof “empty” sets, Russell’s set, the axioms ofZFC, fixed points, Cantor’s theorem, and the possibility of awell-ordering theorem. The Routley set is introduced and studied asa particularly inconsistent object.