By N. S. Hellerstein
This publication is ready 'diamond', a good judgment of paradox. In diamond, an announcement might be actual but fake; an 'imaginary' kingdom, halfway among being and non-being. Diamond's imaginary values resolve many logical paradoxes unsolvable in two-valued boolean good judgment. during this quantity, paradoxes through Russell, Cantor, Berry and Zeno are all resolved. This booklet has 3 sections: Paradox common sense, which covers the vintage paradoxes of mathematical common sense, exhibits how they are often resolved during this new method; the second one Paradox, which relates diamond to Boolean good judgment and the Spencer-Brown modulator"; and Metamathematical drawback, which relates diamond to Godelian meta-mathematics and limitation video games.
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Extra resources for Diamond: A Paradox Logic - Second edition (Knots and Everything)
Xy][z]] = [[[[x]][[y]]][z]] = [[[[x][z]][[y][z]]]] = [[x][z]][[y][z]] ref. trans. ref. Modified Transposition. [[x][yw][zw]] = [[x][y][z]][[x][w]]. Proof. [[x][yw][zw]] = [[x][[[yw][zw]]]] = [[x][[[y][z]]w]] = [[x][[[y][z]][[w]]]] = [[[[x][y][z]][[x][w]]]] = [[x][y][z]][[x][w]] ref. trans. ref. trans. ref. Majority. [[xy][yz][zx]] = [[x][y]][[y][z]][[z][x]]. Proof. trans. reocc. trans.
This topological difference yields a logical riddle which I call the Paradox of the Boundary. The paradox of the boundary has many formulations, such as: What day is midnight? M. M.? Is dawn day or night? Is dusk? Which hemisphere is the equator on? Which longitude are the poles at? Which country owns the border? Is zero plus or minus? If a statement is true at point A and false at point B, then somewhere in-between lies a boundary. At any point on the boundary, is the statement true, or is it false?
If WF is well-founded, then WF is in WF; but this yields the infinite descending element chain · · · ∈ WF ∈ WF ∈ WF. On the other hand, if WF is not well-founded, then any element of WF is well-founded, and element chains deriving from those will be finite. Thus all element chains from WF will be finite; and therefore WF would be well-founded. And so we see that the concept of “well-foundedness” leads us to the paradox of Finitude. Related to “well-foundedness” is “clear-foundedness”. Call a set “clear-founded” if none of its descending chains are more than a Heap of sets deep.