Inconsistent Mathematicswithout a properly developed inconsistent calculus based on infinitesimals, then in consistent claims from the history of the calculus might well simply be symptoms of confusion. This is addressed in Chapter 5. It is further argued that mathematics has a certain primacy over logic, in that paraconsistent or relevant logics have to be based on inconsistent mathematics. If the latter turns out to be reasonably rich then paraconsistentism is vindicated; while if inconsistent mathematics has seri ous restriytions then the case for being interested in inconsistency-tolerant logics is weakened. (On such restrictions, see this chapter, section 3. ) It must be conceded that fault-tolerant computer programming (e. g. Chapter 8) finds a substantial and important use for paraconsistent logics, albeit with an epistemological motivation (see this chapter, section 3). But even here it should be noted that if inconsistent mathematics turned out to be functionally impoverished then so would inconsistent databases. 2. Summary In Chapter 2, Meyer's results on relevant arithmetic are set out, and his view that they have a bearing on G8del's incompleteness theorems is discussed. Model theory for nonclassical logics is also set out so as to be able to show that the inconsistency of inconsistent theories can be controlled or limited, but in this book model theory is kept in the background as much as possible. This is then used to study the functional properties of various equational number theories. |
Contents
ARITHMETIC | 7 |
Relevant Arithmetic | 15 |
ORDER | 39 |
SEVEN THE DELTA FUNCTION | 69 |
Control Theory | 77 |
NINE | 83 |
ELEVEN CATEGORY THEORY | 101 |
Implication | 109 |
THIRTEEN DUALITY | 125 |
147 | |
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Common terms and phrases
assigned atomic sentences axiom background logic calculus category theory Chapter classical consistent complete classical theory closed set complement-topos construction containing d₁ defined Definition Diagonal Lemma differentiation dualises duality equations equivalence classes extensional false finite fixed point method Ft₁ Hence holds iff homogeneous coordinates hyperreal hyperreal number I(Ft₁ I(t₂ identical incomplete theories inconsistent mathematics inconsistent theories induction infinite number infinitesimal integers interpretation intuitionist ISBN k₁ k₂ language Löb's theorem modulo monic multiplication names natural numbers negation nilpotent noninfinite nonstandard nontrivial nonzero open sets operations paraconsistent algebra paraconsistent logics Peano arithmetic pretopology projective geometry Proof properties Proposition provability predicate pullback R-discrete real numbers relevant logic S₁ S₂ set I(t₁ set logic set theory structure t(t₁ t₁ t₂ t2 holds theorem topology topos topos theory toposes transparent true truth predicate truth value WREF zero