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[Longhorn Review] Set Theory and Logic

Material Type: All, Books — Tags: abstract, algebra — Posted on October 3, 2011, 8:44 am

By: Stoll, Robert

A book rather good in two senses, one intentional, the other unintentional. The
intentional sense can be inferred from the Table of Contents. The unintentional
sense is summarized here by ordered pairs of quotations.

I. On the one hand,
after stating Cantor's conception of the term 'set' as any collection of definite,
distinguishable objects ... the author is careful to discuss the words
'distinguishable' and 'definite' as used there. But on the other hand he does not
even hesitate, much less pause, to speak of a set with no elements. How is such a
thing a 'collection of definite, distinguishable objects'? My classroom example was
a box containing glasses - a set of glasses: then take out the glasses, and put them
on the table beside the box - now what constitutes the set of glasses? - is the box
now an empty set of glasses? No, it is a box, the set of glasses is those things on
the table beside the box. Now hit each glass with a hammer, producing pieces of
groken glass - what is the set of glasses? Is there a set of glasses? No, only
pieces of glass. So there is not a set of glasses, and certainly not a null set, or
an empty set, of glasses

II On p.229, the author is careful to phrase the
axiom "for all a in G, ae = ea = a" and the axiom "for each a in G, there exists
..."followed by the canonical comment about omitting the dot denoting the closed
binary operation. But on p.329, the axiom G is "for each a in G, ae = a" instead of
"for all a in G". The item here is not the shift from two-sided identity and
two-sided inverse to left-identity and left-inverse, but rather the shift from "for
all" to "for each" - "for each" in G (both on p.229 and on p.329) clearly means that
each element has its own inverse: the "for each" on p.329 does not mean that each a
has its own identity. The "for all" as distinct from "for each" on p.229 are proper.
Failure to distiniguish between 'for each' and 'for all' is anticipated on p.195
where the author explicitely regards "for every x", "for all x", and "for each x" as
having the same meaning - they do not. Furthermore the author fails to distinghish
between the symbols backward E and backward E!, there exists and there exists
uniquely.

III On p.372 the Bibliographical Note for Section 2 is described as
"a more comprehensive introduction...", but the Note for Sections 3-5 speaks of
"more complete accounts...". Surely 'comprehensive' is a relative adjective and can
be compared by the adverb 'more' - 'more comprehensive' is descriptive. But
'complete' is an absolute adjective - something either is complete or is not
complete - completeness does not have degrees of completeness - 'a more complete
account' is meaningless, a "figment of the imagination" (p.128).

Reviewer: Retired Prof

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