By N. S. Hellerstein
This e-book is ready 'diamond', a common sense of paradox. In diamond, a press release will be precise but fake; an 'imaginary' country, halfway among being and non-being. Diamond's imaginary values clear up many logical paradoxes unsolvable in two-valued boolean common sense. during this quantity, paradoxes by way of Russell, Cantor, Berry and Zeno are all resolved. This publication has 3 sections: Paradox common sense, which covers the vintage paradoxes of mathematical good judgment, exhibits how they are often resolved during this new method; the second one Paradox, which relates diamond to Boolean common sense and the Spencer-Brown modulator"; and Metamathematical quandary, which relates diamond to Godelian meta-mathematics and hindrance video games.
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Additional info for Diamond: A Paradox Logic - Second edition (Knots and Everything)
1H. Finitude H. 21 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. You may replace it with any finite expression. Is Finity finite? If F is finite, then you can replace it by F +1, and thus by F +2, F + 3, etc.
From the bracket axioms we can derive theorems: Reflexion. [[x]] = x. Proof. [[x]] = [[x][[ ]x]] occ. = [[ ][[ ]]]x trans. =x occ. 49 50 Diamond Algebra Identity. [[ ]]x = x. Proof. Directly from Occultation. Domination. [ ]x = [ ]. Proof. , occ. Recall. xx = x. Proof. , occ. Duality. 69 = 9 = 6 =  = [ ] Proof. [ ] = [6] [9]     = = = = comp. , twice [[6[ []]] [[ []]]] trans. [[[[6]]] [[ []]]] trans. , twice [[ ]] =  [6]   [] Ergo  = [[ ]]  = [6]   []   = [6]  [ ]  = [[ ]] ref.
1. In the next chapter, we’ll see that these axioms define diamond logic. Chapter 3 Diamond Algebra Bracket Algebra Laws Normal Forms Completeness A. Bracket Algebra Call these the bracket axioms: Transposition: [[a][b]]c = [[a c][b c]]. [[a]b]a = a. Occultation: Complementarity: [6][9] = [ ]. In addition, we assume commutativity and associativity for juxtaposition: a b = b a; a b c = a b c. These equations are implicit in the bracket notation. Brackets distinguish only inside from outside, not left from right.