By Alexei Davydov, Michael Batanin, Michael Johnson, Stephen Lack, Amnon Neeman

Type conception has turn into the common language of recent arithmetic. This booklet is a set of articles utilising tools of class concept to the components of algebra, geometry, and mathematical physics. between others, this booklet comprises articles on greater different types and their functions and on homotopy theoretic tools. The reader can know about the fascinating new interactions of classification conception with very conventional mathematical disciplines

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**Example text**

The Heap arises at the onset of uncertainty . In practice, the Heap contains a boring number of sand grains ; and the smallest Heap contains the smallest boring number of sand grains! 18 Diamond, A Paradox Logic Finitude. Finite is the opposite of infinite; but in paradox-land, that's no excuse! In fact the concept of finiteness is highly paradoxical; for though finite numbers are finite individually and in finite groups, yet they form an infinity. Let us attempt to evaluate finiteness. Let F ='finitude', or'finity; the generic finite expression .

These normal forms are just like their counterparts in boolean logic, except that they allow differential terms. 46 Diamond, A Paradox Logic Theorem: The Primary Normal Forms F(x) = (A and x) or (B and not(x)) or (C and dx) or D F(x) = (a or not(x)) and (b or x) and (c or Dx) and d where A,B,C,D,a,b,c,d are all free of variable x, and: AorD = F(t) = aandd B or D = F(f) = b and d A or B or C or D = F(i) or F(j) = d D = F(i) and F(j) = a and b and c and d Proof: We get the first two equations from the Disjunctive and Conjunctive Normal Forms by collecting like terms with respect to the variable x.

Students of feasibility will recognize this as a variant of the Boolean Consistency Problem, and therefore NP-complete. Diamond logic's completeness suggest this: Conjecture. Diamond is a "categorical" DeMorgan algebra: Any De Morgan algebra is a subalgebra of images of products of diamond . These De Morgan algebras need not have the Interference axiom; they are subalgebras of ones that do. Thus diamond is to De Morgan algebras as two-valued logic is to Boolean algebra. I consider diamond to be a 2-dimensional extension of twovalued logic that solves paradox, just as the complex numbers are a 2dimensional extension of the real line that solves x2=- 1 .