Das Kalenderblatt 091211

10/12/2009 - 14:23 von WM | Report spam
The difficulty we are confronted with is that ZFC makes a claim we
find implausible. To say we can't criticize ZFC since ZFC is our
theory of sets is obviously to beg the question whether we ought to
adopt it despite claims about cardinality that we might regard as
exorbitant.
[George Boolos: 'Must We Believe in Set Theory?]

Cancer robbed our community of an outstanding philosopher. One is
tempted to say “philosopher and logician,” but as Richard Cartwright
remarked in his eulogy for George Boolos, “he would have not been
altogether happy with the description: accurate, no doubt, but faintly
redundant—a little like describing someone as ‘mathematician and
algebraist.’”

[...] George Boolos made significant contributions in every area of
logic in which he worked. [...] Gödel, in contrast to Boolos, argued
that the axioms of choice and replacement do follow from the iterative
conception. In article 7, an introduction to a posthumously published
lecture by Gödel, Boolos takes issue with Gödel’s Platonistic claim
that the axioms of ZFC (Zermelo Frankel set theory with Choice) “force
themselves upon us as true.” Even if the axioms articulate a natural
and compelling conception of set, they need not correspond to anything
objectively real.

Article 2 contains Boolos’ defense of Fraenkel’s, in contrast to
Zermelo’s, position that first-order but not second-order logic is
applicable to set theory. Boolos criticizes the view of Charles
Parsons (and D. A. Martin) that it makes sense to use second-order
quantifiers when first-order quantifiers range over entities that do
not form a set. Boolos’ answer to the title of article 8, “Must We
Believe in Set Theory?” is ‘no’: the phenomenological argument (due to
Gödel) does not imply that the axioms of set theory correspond to
something real, and the indispensability argument (due to Carnap) that
mathematics is required by our best physical theory, is dismissed as
“rubbish.”

[Gary Mar: "Book Review: Logic, Logic and Logic, George Boolos.
Harvard University Press, 1998. ix + 443 pages. Hardcover $45,
paperback $22.95. ISBN 0-674-53767-X.",
Essays in Philosophy, Vol. 1 No. 2, June 2000
http://www.humboldt.edu/~essays/marrev.html

Gruß, WM
 

Lesen sie die antworten

#1 Peter
10/12/2009 - 21:53 | Warnen spam
On 10 Dez., 14:23, WM wrote:

The difficulty we are confronted with is that ZFC makes a claim we
find implausible. To say we can't criticize ZFC since ZFC is our
theory of sets is obviously to beg the question whether we ought to
adopt it despite claims about cardinality that we might regard as
exorbitant.
[George Boolos: 'Must We Believe in Set Theory?]

In article 7, an introduction to a posthumously published
lecture by Gödel, Boolos takes issue with Gödel’s Platonistic claim
that the axioms of ZFC (Zermelo Frankel set theory with Choice) “force
themselves upon us as true.” Even if the axioms articulate a natural
and compelling conception of set, they need not correspond to anything
objectively real.

Boolos’ answer to the title of article 8, “Must We Believe in Set Theory?”
is ‘no’: the phenomenological argument (due to Gödel) does not imply
that the axioms of set theory correspond to something real, and the
indispensability argument (due to Carnap) that mathematics is required
by our best physical theory, is dismissed as “rubbish.”

[Gary Mar: "Book Review: Logic, Logic and Logic, George Boolos.
Harvard University Press, 1998. ix + 443 pages. Hardcover $45,
paperback $22.95. ISBN 0-674-53767-X.",
Essays in Philosophy, Vol. 1 No. 2, June 2000
http://www.humboldt.edu/~essays/marrev.html



Carsten schrieb gestern:

Was Dein Kalenderblatt so lustig macht, ist, dass Du nicht zwischen
Mathematik, Philosophie und Unfug unterscheidest.



Da ist was dran ;-) Aber wer kann das schon? Auch das will geübt sein,
nicht nur das Wurzelziehen.

Was Dein Kalenderblatt interessant macht, ist, dass Du bisweilen
gute Mathematik und Philosophie darin unterbringst.

Der Hinweis auf George Boolos und dessen Buch "Logic, Logic and Logic"
gehört sicher zu letzterem.

Ich habe jedenfalls die Anregung aufgegriffen, die oben angesprochenen
Artikel 7 und 8 gerade noch einmal durchgelesen, und kann nur sagen:
Wer das Buch von Boolos noch nicht kennt, der sollte es sich zu
Weihnachten
gönnen.

Danke für ein gutes KB.

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