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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GLOSSARY
A Glossary of some of the scientific and mathematical terms that appear in
the PAIAS web site.
Axiom — an axiom is a postulate (proposition or theorem) assumed to be “true” without proof, a primitive statement of a formal deductive system. A formal theory has (formally described) axioms and rules of inference from which are derived the (other) theorems of the theory. Axiom of
Abstraction
— the axiom (technically, an axiom schema) that, given a property P
(where P(x)
indicates that P is true of
x), there exists a set whose members are
precisely those entities having property P.
Formally it is written: An essential problem that is perennially overlooked with this axiom, and the Axiom of Separation, is that P(x) is so general that it opens the door to partial recursion. A partial recursive function PRF(x) is not necessarily well-defined for “"x”; in fact, it is not only not well-defined, it can go off into La-La Land for a given x, never to return (like “Charley ’neath the streets of Boston”). This La-La Land phenomenon, studied somewhat carefully in the Theory of Partial Recursive Functions, is studiously ignored in Logic. Axiom of Choice — (often abbrev. AC) one of the most controversial axioms in Set Theory. There are various, sometimes seemingly conflicting statements of it (not the reason for the controversy), 3 common variants of which are:
It has been proven independent of the other axioms of Set Theory. In 1940 Gödel showed that if ZF without the Axiom of Choice (or Continuum Hypothesis) is consistent, then the system obtained by adding AC and CH to ZF is also consistent. The reason for the controversy is the non-constructive nature of the choice set/function, and the fact that invoking it seems (to some) to lead to uncomfortably more... “paradox”. It is considered equivalent to Zorn’s Lemma, Transfinite Induction, the Well-Ordering Principle/Theorem, and Hausdorff’s Maximality Theorem. It is the view of PAIAS that the Axiom of Choice and the controversy that surrounds it are red-herring type issues. Its use makes it possible to construct sets that hint that there were/are pre-existing problems (as yet unrecognized as fatal ones) in Set Theory (all major variants of Cantor), and that is the reason for the controversy. The Axiom of Choice is a variant of the messenger who gets abused when bringing bad news. But Set Theory would be dull without it, so most people choose to include it (see ZF and ZFC). Axiom of
Separation
— in 1908 Zermelo
introduced his variant of the Axiom of
Abstraction the Axiom of Separation (technically, an axiom
schema; also called the Axiom of Comprehension or Axiom of Subsets): An essential problem that is perennially overlooked with this axiom, and the Axiom of Abstraction, is that P(x) is so general that it opens the door to partial recursion. A partial recursive function PRF(x) is not necessarily well-defined for “("x)”; in fact, it is not only not well-defined, it can go off into La-La Land for a given x. This La-La Land phenomenon, studied somewhat carefully in the Theory of Partial Recursive Functions, is studiously ignored in Logic.
“Axiom of
Separation Paradox”
— Zermelo’s
Axiom of Separation was introduced by him
to avoid paradoxes such as Russell’s
Paradox, a paradox of non-self-membership which is inherent in
the Axiom of
Abstraction. But, unrecognized till now, the Axiom of Separation
has essentially the same paradox (see
Zermelo’s Great... Oversight):
i.e. there is a necessary consequence of the Axiom of
Separation, one that follows
from letting P be the property of non-self-membership, P(x)
= xÏx,
and letting x =
y (forced by ("x)).
Formally it can be written:
Banach-Tarski Paradox — a measure theory paradox where a solid ball is divided up into a small number of pieces that can be moved rigidly so as to form 2 solid balls with the same radius as the original sphere. Some accounts also say that a sphere with twice the radius can also be formed in this way. Their theorem used to prove this result used the Axiom of Choice. Banach and Tarski intended the result to prove that the Axiom of Choice was fundamentally faulty, but it really just popularized the notion that the Axiom of Choice opened up set theory to more of the paradoxes that were inherent in infinity. It is this author’s view that the Banach-Tarski Paradox actually derives from set theory’s transfinite arithmetic result that 2·a0 = a0, which is derived in ZF, i.e. ZFC without the Axiom of Choice. Bijection — a bijection is a mapping from a set onto another set (which may be the same set) such that every element of the from set is mapped onto precisely 1 element in the onto set, which latter is its image, and every element of the onto set is the image of precisely 1 element of the from set. Cantor, Georg Ferdinand Ludwig Philip (1845-1918) — German mathematician, founder of Set Theory, who also made important contributions to classical analysis and topology. Between 1874 and 1895 he developed his theory of transfinite sets, numbers, and arithmetic. Cantor showed that sets could have higher cardinality than that of the set of all natural numbers. Cantor’s Paradox — perhaps the first paradox derived from Cantor’s Set Theory, it was put forward by Cantor himself. If one assumes that there is a set of all sets (or of “everything”), then it contains every subset of itself, and thus its cardinality (or cardinal number) must be greater than or equal to that of its power set (the set of all subsets); but Cantor had already showed that the power set of a set must have a higher cardinality than the set, so a set of all sets must have a cardinality that is both less than that of its power set, and greater than or equal to that of its power set, Cantor’s Paradox. This paradox, like Russell’s Paradox, essentially has something to do with general considerations concerning the definition/construction of sets. This is examined in detail in Russell’s... Oversight. Cardinal
Number
— the cardinal number (or cardinal or cardinality) of a set
is a number that indicates the number of elements in a set, i.e. its
cardinality. The
natural numbers (and zero) are all also finite cardinal numbers
(or equivalent thereto); the first transfinite (infinite)
cardinal number is
a0, the cardinality of the set of
all natural numbers
Á 4 {1,2,3…}.
Perhaps the most fundamental theorem of the arithmetic of transfinite cardinal
numbers is that a0 + 1 = a0.
Cantor showed that there exist yet larger cardinal numbers, e.g.
Cardinality — the cardinality of a set is the cardinal number associated with a set, i.e. its size or the number of elements of the set. If 2 sets can be put into a one-to-one correspondence, then they have the same cardinality. Notationally, the cardinality of a set A is often represented by |A|, n(A), or Ā. Consistency — a theory is consistent if it is not possible to derive, from the axioms and rules of inference of the theory, both a theorem (proposition) and its negation. Continuum
— “the continuum”
is usually held to be the set of real numbers. But one can also refer to
“a continuum”. The
cardinality of the continuum is held to be Continuum
Hypothesis
— (often abbrev. CH) the hypothesis (by Cantor; so-called
because it was unproven; proven by Cohen in 1963 to be independent of
ZF) that the cardinality
of the continuum,
Finite Induction —
finite induction is a bridge from the finite to the transfinite, one that
has been around a long time in terms of the evolution of modern mathematics. It
enables one to give a finite proof for an infinite number of propositions
(sometimes called an “infinite/transfinite schema”), with
each proposition a function of a natural
number n.
Another, standard way of describing it is as a method of proving that all
natural numbers have a certain property, P.
It consists of proving: Frege, Friedrich Ludwig Gottlob (1858-1925) — German philosopher, logician, mathematician; now regarded as a crucial figure in the history of logic and philosophy (and to some extent modern mathematics). Frege and Wittgenstein (whom he influenced greatly) are considered by some to be the source of modern philosophy of language. His axiomatization of Set Theory (from which he believed arithmetic could be derived) met with hostility and rejection. Bertrand Russell, one of his few admirers, found a paradox in Frege’s axiomatization that became famous as Russell’s Paradox. Hilbert, David (1862-1943) — German mathematician, very famous. He developed the concept of a Hilbert Space. Hilbert posed 23 famous questions for mathematicians of the 20th Century, of which ~3/4 have been solved or had significant advances made. Hilbert’s rhetorical question: “What mathematician would want to be expelled from the paradise which Cantor created?” is still quoted. Image — when mapping from a set S1 onto another set S2, if x is an element in S1, the image of x under the mapping m is the element m(x) in S2. The term image sometimes refers to the images of the elements of a subset of S1. Sometimes, instead of sets and elements, the terms spaces and points are used. Inconsistency — a Theory is inconsistent if it is even possible to derive a contradiction from the axioms and rules of inference. I.e. a theory can be inconsistent without anyone being aware that it is, making the question of its Consistency a burning one, at times. Gödel showed that it is essentially impossible to prove the consistency of any theory that contains arithmetic. Mapping — (map) the most general term in mathematics for relating the elements of a set to the element of another set (perhaps the same set). A function is a special case of a map/mapping where each element of a set has only one element of the second set that is its image, i.e. that it maps onto. (Topology uses “mapping” to mean a continuous function.) Natural Numbers —
the natural numbers are the elements of the set of all natural numbers Á 4 {1,2,3...}.
They are defined by a simple variant of
Finite Induction (or
vice versa): Non-Cantorian Set Theory — a (“classical” or “standard”) non-Cantorian set theory is merely an otherwise standard set theory that adds the negation of the Continuum Hypothesis (some say just the Generalized Continuum Hypothesis) or the negation of the Axiom of Choice as an axiom. Paraconsistency — paraconsistent logics and inconsistent mathematics, which have become popular in recent decades (the latter half of the 20th Century), allow inconsistency to exist without then being able to prove all possible theorems. So, for example, they must forgo an “Expansion Rule” which says that for any propositions P and Q, P É (Q É P), i.e. if P is true, then any Q implies P. (A question that is never asked is just where does that “any Q” come from?) In symbolic logic it might look like A ® A Ú 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 — a seemingly logically contradictory situation. A classic example is that there seem to be as many even natural numbers as natural numbers, because for every natural number n there exists the natural number 2n. Before Cantor, paradox used to be considered inconsistency of the mathematically unacceptable kind. But, after some initial dissention, Cantor’s Set Theory was well-received despite the fact that it, and every attempt at its axiomatization since then, have inherently contained many “paradoxes”. Since Cantor’s success (see Hilbert’s rhetorical question) paradox has come to be effectively distinguished from inconsistency of the mathematically unacceptable kind, and the paradoxes inherent in Set Theory are considered to merely “paradoxes of infinity”, i.e. paradoxes that are inherent in any concept of infinity, and thus inescapable in any mathematical theory that contains infinity of any kind. Power Set — the power set of a set S is the set of all the subsets of S. Cantor showed that the cardinality (or cardinal number) of the power set was always greater than the cardinality of the set itself. This led Cantor to the infinite sequence of transfinite cardinalities that is the foundation of the transfinite arithmetic of his Set Theory. Roche Limit — as one gets further away from a (larger) gravitational body, there is a place (not well defined, for various reasons) where the gradient of the gravitational field changes so as to allow (small) masses to stay together by their mutual gravitational attraction alone (abstracting out e.g. electromagnetic and molecular forces). The closer to the larger gravitational body, the more quickly they are torn apart by that gravitational gradient. (It tends to take place slowly, except in the case of e.g. black holes.) For example, the rings of Saturn almost all occur within its calculated Roche limit. Russell, Lord Bertrand Arthur William (1872-1970) — English philosopher, logician, mathematician; one of the most famous such (philosopher, etc.) of the 20th Century, especially noted for his ground-breaking work in mathematical logic and the foundations of mathematics. In collaboration with Alfred North Whitehead, Russell wrote Principia Mathematica (3 vols, 1910-13), an attempt to derive all of mathematics from purely logical assumptions (e.g. they attempted to derive 1+1=2 from Set Theory, and spent quite a few pages of arcane formulae doing so). Russell’s Paradox — in 1902 Bertrand Russell put forward a paradox in the axiomatization of Set Theory proposed by Gottlob Frege that has come to be one of the most paradigmatic paradoxes in modern mathematics. It concerns self-membership and non-self-membership of sets: if we define a set of all sets that are not members of themselves, the paradox is the question of whether this set is a member of itself; i.e. we have an “if it is, then it isn’t, and if it isn’t, then it is” paradox. Russell attempted to avoid this paradox by developing his theory of types. The paradox relates essentially to the Axiom of Abstraction (which see for more formal detail), that many have found fault with and attempted to put right, but it has never been successfully resolved. It also, like Cantor’s Paradox, essentially has something to do with general considerations concerning the definition/construction of sets. This is examined in detail in Russell’s... Oversight. Set — (also called a class, with subtle differences in usage) a collection of objects; the objects are called the set’s elements or members. The collection is an entity in its own right, and a set can have other sets as elements. 2 sets are equal if every element of either of the sets is also an element of the other set; sets can contain infinite numbers of elements. There is no order inherent in the elements of the set, although orderings can be applied to them giving an ordered set. The concept of set has come to be considered fundamental to essentially all of mathematics, i.e., Set Theory is considered, along with Logic, to be the theoretical foundation(s) of mathematics. There is an as yet unrecognized problem in defining a set by specifying its elements (i.e. Russell’s Paradox has not yet been properly resolved). Set Theory — the theory of Sets first put forward by Georg Cantor, and later axiomatized by many famous mathematicians, including Frege, Zermelo (modified later by Fraenkel, giving us ZF), von Neumann (simplified and expanded somewhat later by Bernays and Gödel), and others. Theorem — a proposition proven within a theory from the axioms and rules of inference of that theory. (An axiom of that theory is a theorem that can be considered to be trivially proven by applying the rule of inference equivalent of the an identity operator/element.) Theory — formally, a (mathematical) theory has come to mean the axioms, rules of inference, and all the theorems that can even possibly be derived from them. (Hilbert was a driving force behind this formalization.) A formal language for specifying these is sometimes required. The Consistency/Inconsistency of a theory has historically been one of the most important considerations, followed more recently by the question of its completeness (see Paraconsistency). Universal Quantifier — written “"x” or “for all x”, you see it in various propositions. It is this author’s view that there are overlooked problems with the universal quantifier since it is really equivalent to “for all elements of the set of all things” and shares the “paradoxes” that can be associated with that “set”. See details in “Axiom of Separation Paradox”. Zermelo, Ernst Friedrich Ferdinand (1871-1953) — German mathematician whose Untersuchungen über die Grundlagen der Mengenlehre (Investigations on the Foundations of Set Theory, 1908) formed the foundation of modern axiomatizations of Set Theory (see ZF). ZF — short for Zermelo-Fraenkel, the axiomatization of Set Theory developed by Zermelo and later modifed by Fraenkel (and very importantly Skolem, who was mysteriously left out of the acronym). Along with ZFC, ZF is the most standard and most popular axiomatization(s) of Set Theory. Zermelo developed it very strictly to try to avoid all hint of inconsistency or paradox, especially avoiding constructions such as the set of all sets of Cantor’s Paradox. This later led to the inclusion of the Axiom of Choice to try to get back the less restrictive nature of earlier, less formal, more intuitive approaches to Set Theory. ZFC — short for Zermelo-Fraenkel-Choice, i.e. ZF with the Axiom of Choice.
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, the
cardinality of the set of real numbers. By definition, the next largest
transfinite cardinal number after
