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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Set Theory has serious problems with theoretical consistency. Fundamental... Oversights have affected the ability of mathematicians to detect actual inconsistency — the bad kind, as opposed to “Oh, that! That’s just a paradox.” (None of the classical-standard set variants of theory — ZF, ZFC, von Neumann, etc. — is “paraconsistent”. A paraconsistent logic is one in which the theory may be inconsistent with regard to a particular theorem and its negation without proving every possible theorem. Paraconsistency has become a popular study in philosophy in recent decades.) For example, when the question of set theory’s consistency comes up, most mathematicians forget what the formal definition of the inconsistency of a theory actually is. It may sound unbelievable to some, but they will say things like “the assumption that you can validly derive that result is false, as evidenced by the fact that you get a contradiction.” They will say this even when the “assumption” in question is inherent in the axioms and rules of inference of the theory in question. (This is at the very least a somewhat frightening theoretical... oversight, since if one is never allowed to derive a contradiction then any theory whatsoever becomes trivially “consistent”.) Another example is that mathematicians also overlook that one must always be able to replace a defined entity by the definition. One mathematician — a reviewer for a well known mathematical association — said [paraphrasing and accenting]: your proof is invalid since it refers to the serial-sequential construction-definition of the involved entities; these entities now have a ‘simultaneous existence’. (See Fundamental... Oversights for more on all this.) The first approach we offer to exploring the problem of the inconsistency of set theory more deeply is an unusually lighthearted (given the subject matter), even humorous one. Even non-mathematicians can appreciate the essentials.
The Good Shepherd’s Paradox QUICK FORMALITY: But for those who want formality, here is a very quick but more formal description of (part of) the problem (for a rigorous formal and far more detailed approach, see Bijection Paradoxes):
Although set theory was once considered by many to be too paradoxical, i.e. fatally inconsistent, they were silenced by the overwhelming brilliance of Cantor, Dedekind, Hilbert, and many others. Today, set theory is so accepted that it is no longer even considered a field of study in mathematics. One must specialize greatly within and-or beyond set theory to get even a major field of study. The paradoxes that still pervade the transfinite portion of set theory and that once threatened to sink our ship, so to speak, are now considered quite different from, and handled very differently from, formal (fatal) inconsistencies, of which no one breathes even a mention in conjunction with the foundational mathematics of set theory. This will all change, hopefully soon.
A More Detailed Introduction to Set Theory and Its InconsistencyThe set theory developed by Cantor (Georg Ferdinand Ludwig Philip, 1845-1918) has been paradoxical since its inception. In fact, with the acceptance of Cantor’s set theory, paradox, which had earlier meant inconsistency in a mathematically and logically unacceptable sense, has come to be considered not merely acceptable, but conventional. The acceptance by mathematicians of this marriage of paradox and consistency in the two thirds of modern mathematics that has Cantor’s set theory — along with logic — as an essential part of its foundations can be summed up in David Hilbert’s rhetorical question: “What mathematician would want to be expelled from the paradise which Cantor created?” In order not to be “expelled”, i.e. in order to avoid turning acceptable paradox into unacceptable inconsistency, we even selectively abandon fundamental principles of mathematics. A trivial — and damning — example in set theory is the avoidance of the otherwise standard application of the standard rule of inference that equal quantities can be subtracted from both sides of an equation to the fundamental equation of transfinite cardinal arithmetic: a0 + 1 = a0. There are many more such examples. This result — that a0 + 1 = a0 — is often proven by means of constructing a bijection from Á ~ {0} onto Á, demonstrating that |Á ~ {0}| = |Á| (where |S| is the cardinality of the set S, and Á 4 {1,2,3…} is the set of all natural numbers, and thus the set union of Á and {0} has 1 more element in it than Á). (It can be noted in passing that modernly some people have started defining Á as Á 4 {0,1,2,3…}, but this should not be a problem here.) Since |Á ~ {0}| = |Á| 4 a0, and |Á ~ {0}| = |Á| + |1| = a0 + 1, the equation a0 + 1 = a0 is considered to follow immediately. (Eventually we will offer: See the section “‘Counting’ and Cantorian ‘Reordering’”, for an analysis of this ostensibly bijective mapping from Á ~ {0} = {0,1,2,3…} onto Á 4 {1,2,3…}.) In the face of a0 + 1 = a0, since the theorem 1 = 0 would be considered not merely paradox but inconsistency (unacceptable since e.g. all natural numbers would become provably equal), we back off from the usual definition of a mathematical theory being all the theorems that can possibly be derived from the axioms and rules of inference, and from the usual definition of the inconsistency of a theory being the possibility of formally deriving both a theorem and its negation from those same axioms and rules of inference. Relatedly, we also back off from the fundamental principle of always being able to replace a defined/constructed entity/symbol by its initial definition/construction. E.g. once we define/construct transfinite arithmetic from finite arithmetic, we allow proofs using a “simultaneous” mapping from n to n + 1 for all natural numbers n in Á 4 {1,2,3…} without further reference to the serial definition/construction of the natural numbers and thus the mapping, and we get very different results. And we back off from these formal definitions and principles for the wrong reasons. (See more detail on these highly questionable practices in Fundamental... Oversights in Mathematics.) With regard to the fundamental theorem of set theory that a0 + 1 = a0 we have selectively, in a dubious context sensitive fashion, deliberately failed to apply a standard rule of inference merely because it would yield a standard inconsistency (in a theory not noted for its paraconsistency; see below). We can speak of hidden “rules of deference” in such situations, where we quasi-formally refrain from performing a standard derivation, e.g. applying a standard rule of inference, which derivation/application, by definition of a mathematical “theory”, gives us a valid theorem of the theory, and gives us a contradiction if and only if the theory is inconsistent. Any mathematical theory will be “consistent” if we proceed on such a basis. (Paraconsistent logics and inconsistent mathematics, which have become popular in recent decades, allow inconsistency to exist without then being able to prove all possible theorems. So, for example, they might forgo an “Expansion Rule” which says that for any propositions P and Q, P > (Q > P), i.e. if P is true, then any Q implies P; in symbolic logic it might look like A → A ? B. Once one obtains both P and not P, i.e. the first contradiction, it is then trivial to derive any not Q, and thus any proposition Q. Similarly for symbolic logic.) Paradox in set theory is usually associated with the Axiom of Choice, the use of which seems to lead to rather more paradox than many mathematicians are comfortable with. But, in fact, unacceptable paradox in set theory will be seen to derive from the Continuum Hypothesis, or rather from the fundamentals that give rise to it. It is the assumption (ostensibly proven) that transfinite sets can be bijected with proper subsets of themselves that leads visibly to the Continuum Hypothesis, and invisibly to “problematic” paradox. Besides “paradoxes” (such as above) that are known but not yet adequately appreciated, there are many new paradoxes that are as yet unrecognized in set theory and related theories such as real number theory. These paradoxes, some of which will be studied in later sections, also suggest that a new kind of non-classical non-Cantorian set theory is needed for transfinite sets and their arithmetic, at least one that satisfactorily resolves all known paradoxes, especially the more embarrassing ones, in particular the new ones presented in this paper. “Non-Cantorian set theory” has been defined, with great lack of imagination concerning the entire range of possibilities, as an otherwise standard set theory that axiomatically rejects (as opposed to deriving a falsification of) the Continuum Hypothesis (see definition), which has so far seemed to be an independent axiom (e.g. Gödel and Cohen). However the Continuum Hypothesis will be found to be not only falsifiable, but provably false (see Theorem 12 in Bijection Paradoxes), and a new set theory will be needed that more than just rejects it, or even falsifies it (since it is derivably false within standard, non-paraconsistent set theory). An alternative to the Continuum Hypothesis, a “Quantinuum Hypothesis”, will be seen to derive naturally from all this, and to suggest paradigms for a new theory. The desirability of developing a Quantinuum Hypothesis and a new and non-standardly non-Cantorian set theory (or theories) will be proposed in these web pages, but in the sense of necessity, not just in the sense of the explorations of new systems for their own sake as with non-Euclidean geometries. A new theory is needed with a whole new concept of infinity or “transfinity”, one from which e.g. “quantinuous” infinitesimals / “transfinitesimals” derive naturally. An example of a possible benefit of such a new non-Cantorian theory is that it might offer a resolution to the problem of renormalization that we find in e.g. quantum mechanics, where, related to the a0 + 1 = a0 paradox, infinite quantities are subtracted — with a hope and a prayer — from both sides of various equations to get equations of finite quantities that are theoretically useful.
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