[FOM] From theorems of infinity to axioms of infinity
Nik Weaver
nweaver at math.wustl.edu
Sun Mar 17 13:36:12 EDT 2013
Mart Dowd wrote:
> Regarding the power set axiom, while it is true that classical
> mathematics can be carried out in the first few levels of the cumulative
> hierarchy, calling set theory a "metaphysical extravagance " seems
> extreme.
I actually have a principled reason for saying this.
What is the difference between a set and a proper class?
We have two ways of specifying a collection. One is by stating a
condition that allows us to recognize whether any given object does
or does not belong to the collection. The other is by explicitly
listing all the elements of the collection.
We know that some collections cannot be explicitly listed out, even in
principle. We could never enumerate all sets. Any explicit list of
sets would itself be a set and could then be added in to enlarge the
list. Michael Dummett calls this phenomenon "indefinite extensibility".
My view is that sets are collections which could, in principle, be
explicitly enumerated --- I call this property "surveyability" --- and
proper classes are collections which can only be specified indirectly,
by stating a criterion for membership.
Now there are a range of views as to what is possible in principle.
Ultrafinitists draw the line at "feasable" computations and would
consider the numbers between 1 and 10^10 as constituting what I
would call a proper class, although they would certainly agree that
the concept "a string of ten digits" is not ambiguous. Finitists
consider any finite collection, but no infinite collection, as being
surveyable in principle. I personally am more sympathetic to this
view than I am to ultrafinitism, although I do think that surveying
infinite collections can be possible in principle, even if it is not
(presumably) physically possible in our world.
The power set axiom is a metaphysical extravagence because it pushes
the notion of surveyability from countable collections like the
natural numbers, where we have a clear intuition of their sequential
availability, to uncountable collections like the power set of omega,
for which we have no such intuition and have not the slightest idea
of what it would be like, or how we could go about, making an
exhaustive survey.
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"?
Nik
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