[FOM] From theorems of infinity to axioms of infinity
MartDowd at aol.com
MartDowd at aol.com
Wed Mar 20 13:14:23 EDT 2013
Nik,
J. Shoenfield's article in the "Handbook of Mathematical Logic" contains a
justification of the power set axiom. Its adoption can be seen as an
application of Occam's razor. It is necessary to have the set of all functions
from the reals to the reals, even for calculus; then subsets such as the
continuous functions,and differentiable functions can be defined. This
illustrates that the power set of a set is a self-evident and useful concept;
the simplest way to ensure its existence in formal set theory is to adopt the
power set axiom. The fact that this leads to sets outside the realm of
ordinary mathematics could be considered a metaphysical accident rather than
an extravagance.
= Martin Dowd
In a message dated 3/19/2013 5:24:01 P.M. Pacific Daylight Time,
nweaver at math.wustl.edu writes:
The point I have to emphasize is that I am not aware of any even
remotely cogent justification for making this jump. There is no
clear philosophical reason to suppose that infinite power sets are
surveyable, nor is there any practical reason, as it turns out that
the vast bulk of mainstream mathematics can do perfectly well without
this assumption. Does that help explain why I use the term "metaphysical
extravagence"?
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