By N. S. Hellerstein

This ebook is set "diamond", a good judgment of paradox. In diamond, an announcement will be actual but fake; an "imaginary" nation, halfway among being and non-being. Diamond's imaginary values resolve many logical paradoxes unsolvable in two-valued boolean common sense. Diamond is a brand new technique to clear up the dilemmas of upper arithmetic. during this quantity, paradoxes through Russell, Cantor, Berry and Zeno are all resolved. This booklet contains sections: effortless; which covers the vintage paradoxes of mathematical common sense and indicates how they are often resolved during this new method; and complicated, which relates diamond to Boolean common sense, three-valued common sense, Gödelian meta-mathematics and problem video games.

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Completeness and Categoricity The differential normal forms imply this theorem: Completeness. Any equational identity in diamond can be deduced from the diamond laws. Proof: By induction on the number of variables. Let F = G be an identity with N variables. ) If N = 0, then F = G is an arithmetic equation. Since the diamond laws imply diamond's truth tables, F = G follows from those axioms. ) Suppose that all N-I variable identities in diamond are provable from the diamond laws. Let F(x) be F considered as an expression in its Nth variable x.

In "phased delay", one permits a standard switching circuit to oscillate. T then means "on", F means "off', and the two mid-values mean the two wobbles of opposite phase. This implementation requires a "global clock" to ensure that the switches wobble in synchrony; it also reqires that all inverter switches take a unit delay, and all "positive" (and, or, majority) gates take delay two: (not B ) (n) = not ( B(n-1) ) (A and B) (n) (A(n-2) and B(n-2) ) (A or B) (n) _ (A(n-2) or B(n-2) ) (M( A, B, C)) (n) = (M( A(n- 2), B(n-2), C(n-2) ) In a "dual rail" circuit, all wires in a standard switching circuit are replaced by pairs of wires.

1 F. " If this defines a number, then it has done so in only nineteen syllables, and therefore is its own successor . Presumably this number = Finitude, if your boredom threshold is twenty syllables. 20 Diamond, A Paradox Logic G. Game Paradoxes. Hypergame and the Mortal. Let "Hypergame" be the game whose initial position is the set of all "short" games - that is, all games that end in a finite number of moves. For one's first move in Hypergame , one may move to the initial position of any short game.