Log24

From page 180, Logicomix —

Alfred North Whitehead in the first of
the above-named years, 1906 —

"But the project's central problem was always there."

"The deeper we got into our Quest…
…The more I doubted its premises."

— Attributed to Bertrand Russell
by Apostolos Doxiadis and Christos
Papadimitriou in Logicomix  (2008-9)

For some background, see "Cartoon Graveyard" and "Many Dimensions."

Mathematics and Religion, continued–

Calvin Jongsma, review of an anthology titled Mathematics and the Divine—

"Believers of many faiths have found significant points of contact between their religious outlooks and mathematics. Not all of these claims were made in the distant past or by certified crackpots…."

Edward Nelson in "Warning Signs of a Possible Collapse of Contemporary Mathematics"–

"The most impressive feature of Cantor’s theory is that he showed that there are different sizes of infinity, by his famous diagonal argument. But Russell applied this argument to establish his paradox: the set of all sets that are not elements of themselves both is and is not an element of itself."

Jongsma's assertion appears to be true. Nelson's appears to be false. Discuss.

Remarks:

Saying that someone applied some argument– any argument will do here– to establish a paradox– any paradox will do here– casts into doubt the validity of either the argument, the application of the argument, or both. In the Cantor-Russell case, such doubt is unnecessary, since the paradox is clearly independent of the diagonal argument. There is certainly an historical connection between Cantor's argument and Russell's paradox– see, for instance, Wikipedia on the latter. The historical connection is, however, not a logical connection.

For Russell discovering his paradox without the use of Cantor's diagonal argument, see Logicomix—

Click for some context.