[FOM] Why not this theory be the foundational theory of mathematics?
Thomas Klimpel
jacques.gentzen at gmail.com
Wed Mar 27 08:21:55 EDT 2019
Dear Zuhair,
you ask "Why not this theory be the foundational theory of mathematics?"
I would say it is simply not sufficiently different from ZFC. Remember
that reference to ZFC implies the possibility to add some inaccessible
cardinals if necessary. For example, HoTT with propositional
truncation and choice on the level of sets needs two inaccessibles,
and this is not considered to be a big deal. But HoTT itself is so
different from ZFC in its basic approach that it is worthwhile to use
it as an alternative foundation.
Also category theory would be very different from ZFC, but it is not a
foundational theory. It is not, because it is too vague, and
implicitly uses higher order quantifiers. (The category of all sets is
specified by a sort of "last word quantification", but this is
troublesome for a foundational theory. A foundational theory should
lay down the first words, not insist on having the last word.)
> The following is a link to clearer exposition of this theory:
I don't see how this should be clearer than your initial email. It
contains fewer information and references, and doesn't explain
anything not explained in your initial email.
> All axioms of class theory and set theory can be interpreted in a first order
> theory (with equality) plus primitives of set membership "\in", and "W" standing
> for some fixed set.
I remember having seen the use of an explicit constant for the class
of all sets before. I just checked some references, but failed to find
it again.
> This theory also fulfill all of F.A. Muller's criteria for a founding theory of mathematics!
> for Muller's criteria see:
> http://philsci-archive.pitt.edu/1372/1/SetClassCat.PDF
Muller's article also uses an explicit constant for the class of all
sets, but he uses "V" instead of "W". Assuming you based your theory
on the description from Muller's article, why did you replace "V" by
"W"? How is your axiomatic theory related to the axiomatic theory
presented in Muller's article? Is it the same? Or is it more powerful?
> 2. Set Comprehension schema: if \phi is a formula in the pure language of set
> theory [i.e., doesn't use the symbol "W"], then:
>
> x_1,..,x_n \in W -> [\forall y (\phi -> y \subset W) -> \forall y(\phi -> y \in W)],
>
> is an axiom.
>
> In English: any pure set theoretic formula from parameters in W, that only
> holds of subsets of W; also only holds of elements of W.
I don't remember having seen this axiom scheme before, but I may be
wrong. Is this your invention, or is this basically equivalent to the
"Axiom of Completeness (Compl). The class of all sets V is complete,
i.e. all classes contained or included in sets are sets."
from Muller's article (or some axiom scheme from another reference)?
> Why wan't such a simply presented natural theory qualify as the foundational theory of mathematics?
We already know that there cannot be a perfect foundational theory of
mathematics. ZFC serves the role as a foundation of mathematics well
enough for the moment. The described axiomatic theory is not really
that different from ZFC, with respect to foundation. It may be
different with respect to class and set theory. So studying it as an
alternative set theory could be interesting. In fact, there are many
alternative set theories, and their study is a valid and interesting
field of mathematics.
Regards,
Thomas
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