[FOM] Foundational Challenge

[Michael Lee Finney]

> Therefore I issue the following challenge to proponents of any kind of
> “alternative foundations” to “ZFC plus large cardinals”.

> I CHALLENGE YOU TO IDENTIFY:

> An EXAMPLE of mathematics done in any “alternative foundation” to set theory, where EITHER
> a) it is not obvious that any result derived that is statable in set theory
> will be provable in ZFC plus a large cardinal of some kind

> — JS

That is not the purpose of alternative foundations. Generally, alternative
foundations seek to either

1. Restrict certain modes of reasoning for one reason or another. This may and
usually does eliminate some classical results. For example, what happens if
contraposition is no longer allowed in a logic? What can still be proven? Or
what be proven if choice is no longer allowed? This helps to understand the
structure of mathematics itself, as opposed to using mathematics to understand
the structure of some other object. Not everyone accepts classical
propositional logic as providing valid reasoning under all circumstances.
After all, if it did, would there have been paradoxes in the first place?

2. Provide a better foundation for mathematics. This may or may not increase
the theorems reachable by a mathematician, or may change them in some way. For
example, what happens when the foundation axiom is replaced? Aczel created a
version of set theory that is consistent if and only if ZFC is consistent, but
allows a set to be a member of itself. How does mathematics change under those
circumstances? Do we get new, interesting theorems? Or, another goal would be
to unify ZFC and Category Theory better. How can that be done? Category Theory
is hampered by the size constraints of ZFC. So, to try to make it fit, you add
classes, then superclasses, etc. And you twist the structure of Category
Theory because now you have essentially the same things at different "levels".

3. Provide a completely different foundation of mathematics. Perhaps there is
some alternative foundation that would be better than ZFC. It could even
subsume ZFC. But that doesn't necessarily mean new or different theorems in
most classical areas.

4. Simply to study the structure of some part or all of mathematics. I am not
up on this, but I believe that is the point of Reverse Mathematics.

5. Apply a different type of constraint to the underlying logic. For example,
certain computational processes can be designed to not consume energy by not
destroying information during the computational process. That may require
alternative foundations. It might not lead to new results, but might require
different methods of proof. But, it could potentially make computers much
faster.

6. Then there are alternative foundations such as Quantum Logic that may not
be directly usable for conventional mathematics at all. But might be useful
when building quantum computers that could then be used for conventional
mathematics.

ZFC is successful precisely because most, if not all, areas that mathematics
are interested in, can be modelled in ZFC. But, ZFC was the result of
restricting mathematics in the first place due to the paradoxes. There were
alternative contenders such as von Neumann's set theory, but those are largely
equivalent to ZFC. But, perhaps there is a better foundation than ZFC. Perhaps
more of Cantor's lost paradise can be regained.

Large cardinals may not, and usually don't, play a direct role in alternative
foundations. Alternative foundations is a very rich and interesting area of
research. We still don't have a foundation that restores all of Cantor's
paradise. We have just cleaned up part of the garden.

Michael Lee Finney