[FOM] re the meaning of set

[joeshipman@aol.com]

Lance:

>I'm actually skeptical that one needs a theory of meaning as
>background to a discussion of ordinary uses of "set".

It is going to be difficult to describe the "ordinary" uses of the word 
"set", since according to several sources I have found, that word has 
the longest entry in the Oxford English Dictionary. My Second Edition 
of the Compact OED uses 25 pages for "set"; I cannot independently 
verify Wikipedia's assertion that the OED descrbes over 430 senses of 
the word, or that the entry consists of approximately 60,000 words, but 
it is a monstrously long entry.

The first usage relevant to mathematics appears at the very end of page 
1720i (that is, the last of the nine pages photo-reduced onto page 
1720, originally numbered page 53 in Volume 15 "SER-SOOSY"). It begins:

    10. a. A number of things grouped together according to a system of 
classification or conceived as forming a whole.

The first citation is from Locke 1690, the first in which the word is 
spelled with a single "t" is from Swift 1738.

After many more citations (Carlyle 1890 "The Constitution, the set of 
laws...that men will live under", Cayley 1890 "Let L denote a set of 
any four elements a, b, c, d", etc.), it continues:

   b. Math. Used variously, as defined by the individual author (!!)

Citations begin with Hamilton (1837). The definition eventually 
continues:

    c. Math and Logic.  An assemblage of distinct entities, either 
individually specified or which satisfy certain specified conditions.

The citations here are illuminating:

1857 Phil. Trans. R. Soc. CXLVII 717 Any values (x1, y1, z1,...) 
satisfying the equations, are said to constitute a set of roots of the 
system.

1897 W. Burnside, Theory of Groups. Let a1, a2, ..., a_n be a set of n 
distinct letters.

1903 Trans. Amer. Math. Soc. IV, 27. A set of elements in which a rule 
of combination O is so defined as to satisfy the following three 
postulates shall be called an Abelian group with respect to O.

1937 Jrnl. Symbolic Logic II. 66. According to the leading idea of the 
von Neumann set theory we have to deal with two kinds of individuals, 
which we may distinguish as sets and classes. The distinction may be 
thought of in this way, that a set is a multitude forming a proper 
thing, while a class is a predicate regarded only with respect to its 
extension.

1965 Patterson & Rutherford Elem. Abstr. Algebra i. 3. If s is an 
element of a set S, we write s /epsilonsymbol S.

1972 A.G. Howson Handbk. Terms Algebra & Anal. ii. 8. A set is a 
totality of certain definite, distinguishable objects of our intuition 
or thought -- called the elements of the set. This classic definition 
of a set was given by Georg Cantor in 1874. Such attempts to give 
elementary defintions of a set are, however, doomed to failure, their 
being in the main based on the use of undefined synonyms, such as 
"collection", and leading to logical inconsistencies (see Russell 
paradox...)  For this reason, mathematicians now regard the notion of a 
set as an undefined, primitive concept.

1975 I. Stewart Concepts Mod. Math iv. 47. There is only one empty set. 
all empty sets are equal.


The other usage relevant to mathematics begins in the middle of page 
1721a, and is the final part of the definition of set as a NOUN (there 
follow many more pages about "set" as a verb).

    III. 15. Special Comb: set theory, the branch of mathematics which 
deals with sets without regard to the nature of their individual 
constituents; an axiomatization which allows of the discussion of sets; 
set-theoretic, -theoretical, adjs., of or pertaining to set theory, 
hence set-theoreticlly adv.

Citations follow from Mendelson, Suppes, Kleene, Quine, and Chomsky.

-- Joe Shipman