Palo Alto Institute for Advanced Study
2007-12-18
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Palo Alto Institute for Advanced Study 2007-12-18
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The Good Shepherd’s Paradox (cont.) Final Analysis of The Good Shepherd’s Paradox(If, at any time, you find yourself saying “you can't do that BECAUSE you get a contradiction or something undefined or [any variant thereof]...”, please refer to Fundamental... Oversights.) The... oversights involved in the inconsistency of set theory are manifold (in the general as opposed to the mathematical sense). Please, refer to Fundamental... Oversights for the more general aspects of these. The particular... oversight, the completely crucial one, can be summed up as... oversighting the absolutely essential and inescapable relationship between set subtraction and cardinality. The bijection (in modern terms) that must exist, the strict 1-to-1 (and, remember the old-fashioned term “onto”?) relationship between the members or elements of 2 sets of the same cardinality, is the key to the downfall of set theory. The Golfer’s Paradox and the Banker’s Paradox make it much easier to see, but the essential is that any element that is common to both of 2 bijected sets may as well be mapped 1-to-1 to precisely itself as to the precisely 1 element that it is mapped to. These elements are then effectively “subtracted out” of any further relative cardinality considerations, leaving (ostensibly) bijected only those subsets of elements not common to both sets. This holds even for transfinite sets. If we look at the ostensible bijection of a set with a proper subset of itself — commonly taken as even a defining property of a transfinite set — we can switch the particular 1-to-1 mappings so that all of the elements common to both sets are mapped to each other. That leaves only a non-empty set (ostensibly) bijected with an obviously empty set. The (ostensible) possibility of bijecting a set onto a proper subset of itself has been considered a standard defining property of a standard transfinite set (from Dedekind and, before him, Peirce); this property is inherently contradictory and inconsistent, per se. Pursuing this to the already marked greater generality, we can show:
The proof is trivial, especially after the examination we have already made of The Good Shepherd’s Paradox. Switch the 1-to-1 sub-mappings of each element not common to both sets so that each maps 1-to-1 with itself, and delete these “identity” subbijections from the original bijection from S1 onto S2. This leaves only S1 - S2 bijected onto S2 - S1. The case of a bijection from a set onto a proper subset of itself is now merely a special case. There are many, many, many such “theorems”, ones that would show over and over again that set theory is inherently, standardly inconsistent. And to some extent it will be good to thoroughly analyze many of them. But that is beyond the scope of this popular yet hopefully “sufficiently rigorous” introduction to the inconsistency of set theory, The Paradox of The Good Shepherd, for whom one sheep cannot truly be lost, not in a transfinity of sheep, not even in an absolute transfinity of sheep, nor of anything else.
The Good Shepherd...
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