[FOM] Prof. Martin's "Philosophical Issues about the Hierarchy of Sets"
Marc Alcobé
malcobe at gmail.com
Fri Apr 29 07:34:26 EDT 2011
Dear FOMers,
Some months ago (October 2010), in the context of the Workshop on Set
Theory and the Philosophy of Mathematics, Professor Donald A. Martin
gave a talk entitled "Philosophical Issues about the Cumulative
Hierarchy of Sets":
-- Begin quote --
Abstract: I will discuss some philosophical questions about the
cumulative hierarchy of sets, its levels, and their theories. Some
examples:
(1) It is sometimes asserted one cannot quantify over everything. A
related assertion is that each of our statements about the universe of
sets can from a different perspective be seen as a statement about
some Va. Thus the class-set distinction is really a relative one.
Does this make sense? Is it right?
(2) Is the first order theory of V determinate? Does every sentence
have a truth value? Are there levels of the hierarchy whose first
order theories are indeterminate? If so, what is the lowest such
level? What about L and the constructibility hierarchy?
(3) There are many examples of proofs of a statement about one level
of the hierachy that use principles about a higher level. Under what
conditions and in what sense do these count as establishing the lower
level statement?
I will discuss these questions mainly from a viewpoint that takes
mathematics to be about basic mathematical concepts, e.g., those of
natural number, real number, and set.
-- End quote--
I am highly interested in learning how these questions might be
answered, so I would be grateful if anyone could give any information
in this respect, especially for those questions of 1 and 3 (I am
afraid it is impossible to make justice to 2 in a few lines). I have
also posted this question in MO (just in case anyone wishes to write
something there).
Thank you in advance.
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