[FOM] Standardization of sets, sorts, and free logic
Anthony Coulter
fom at anthonycoulter.name
Mon Oct 19 22:13:47 EDT 2015
In light of the recent discussion about whether N would be a subset of R
in an good, standardized formalization of regular mathematics, I would
like to ask a few related questions. In the below, the "Standard
Library" is whatever ideal formal system the reader chooses to imagine.
1. Should the Standard Library make use of the extended real line?
When analysis books discuss the limit supremum, the limit infimum,
and measure theory, they treat positive and negative infinity as
"physical objects." (Contrast with ordinary limits, where the infinities
are usually abstracted away so they don't appear in the definition at
all.) The trouble with these objects is that they are a bit nastier to
construct---Cauchy sequences don't converge to infinity---and that might
cause problems with some foundational systems.
2. What background set theory is needed to discuss topology and abstract
algebra?
Many different set theories exist (ZF, Kripke-Platek, New
Foundations, etc.) but algebra and topology textbooks tend not to bring
them up by name. The axiom of choice is the only axiom that tends to
warrant special note in such books. Other variations (which seem to be
designed to trim the constructable hierarchy) don't seem to matter to
regular mathematicians. What set theory best captures this background
that mathematicians use? (I personally think the answer to this is
higher-order logic, but I raise the question anyway.)
3. How should the Standard Library deal with the category of all groups?
Saunders MacLane only mentions ZFC in "Categories for the Working
Mathematician" in an appendix, where he mentions it only to show that it
isn't even necessary for the development of category theory. Meanwhile,
it has been claimed that Mizar is based on Tarski-Grothendieck set
theory rather than ZFC to provide arbitrarily large universes for
category theory. How much set theory is actually needed here? (Questions
2 and 3 could be merged, but I separate them because I've read so many
articles bemoaning the restriction to small categories.)
4. Regarding many-sorted logic: How do mathematicians approach types in
actual practice?
Generally speaking, they seem to reject "3\in 5" as a meaningless
ill-typed statement (unless they're dealing with von Neumann ordinals).
But I think they're less likely to reject "1/0" as ill-typed---it's
undefined, but most folks I've talked to consider it a well-formed
expression. But then I wonder: Would a regular mathematician regard "the
sum of 1/n^2 as n goes from e to pi" as ill-typed (like "3\in 5") or
undefined (like "1/0")? [This question gets to the heart of the N\subset
R debate from earlier.]
4. How do mathematicians who are *not* foundations of math people regard
undefined terms and free logic?
In actual practice, the most common approach is to refrain from
proving theorems about 1/0. But the Standard Library needs a Standard
Logic, so "(1+1)/0 = 2/0" and "1/0 + 2 = 2 + 1/0" must each either be
provable or not provable. What conventions would be least offensive to
mathematicians? (My answer: a = b -> f(a) = f(b) is logical but
commutativity is mathematical [and doesn't even apply to all supersets
of R, e.g. the quaternions] so the former is true and the latter is
false.) Likewise, do non-FOM mathematicians think of explicit existence
predicates as natural concepts or ugly hacks (the way we see setting 1/0
= 0)? (I personally think existence predicates are fine.)
Regards,
Anthony
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