I think the blue-eyed islanders problem [1] suggests a solution for replacing sets with categories as the foundations of mathematics. My reasoning is as follows:

1. Non-existance is not a well-defined concept if in isolation. It can only have meaning as attached to that which exists.

2. To exist is to be possible. To not exist is to be impossible. Non-existance is a parasitic concept. Therefore, everything must start with one empty set, which necessarily exists.

3. Once something exists, two things exist. The set, and that which is not it.

4. Two sets exist implies three sets exist, etc.

5. The empty set is always attached to the total set. They envelop each other. That's where motion comes from. A new quanta of energy is added for every new set.

6. 3, 4, 5... this is the expansion of the universe in the category of sets.

7. Sets are both physical and mathematical. The paradoxes from the math manifest as an inability of the maximally expanded universe to channel the energy flow (motion ceases for one frame). The rate of energy expansion (set expansion, mathematically) exceeds the capacity of the current Universe topology to accomodate it (and I don't know enough to understand how that happens), which leads to symmetry breaking (fractalization). This causes a "stepping up" of the game (it's literally a step change). After the symmetry breaking, the matter in the region of the new fractalic branch is in the next higher topology. For us, it means we have transitioned to matter with categories replacing sets (this adds 1 bit of information to the existing Universe).

8. It seems the Universe enumerates all states that are possible up until the energy flow limit occurs. They are by definition finite, so they will cycle. They must be finite, but for a reason I am not qualified to understand. I only believe it because I can count several times this has already happened in the past, so it must continue to happen. Handwaving, it should be possible to prove that the energy flows (and I don't quite know what that means really; I think it means bits) will always exceed the expansion that happens in the current topology, making the next symmetry breaking necessary. This is the part that I don't know if it makes sense.

9. After enumerating all possible states in one topology, the flow cracks the fabric of the fractal to start a new zoom level with the next higher topology, adding one bit of information and room to swirl into (one bit is enough for the shuffle to work).

10. The Universe explores all possible states adding one bit for each fractalization. These fractalizations are new Big Bangs embedded in the expansion of the previous Big Bang, which now expand into the higher topology, until the new flow capacity is exceeded.

11. Etc.

The consequence of this is that once the fractalization occurs, the game is fundamentally changed. Moebius strips of energy flows can now be broken, whereas in the set topology they are necessarily there because of the liar's paradox. Coincidentally, this is Buddha's statement:
- to be is to not be (1)
- to not be is to be (2)

This cannot be otherwise because the set is not yet attached to the nothing. It is that addition to the structure (1 bit of information) that allows the expansion to continue, because now there is more empty space available, just enough to start another shuffle. Of course, only in the deepest region of the fractal does the new topology exist. In this case, that space would be in our own brains, which are predicated on sets (on a fundamental physical level).

12. Liar's paradox turns into a tautology in the world of categories:
- the meaning of to be is the meaning of to not be (3)

Which just says that a new bit of information is attached to every set, after which we identify the yin with the yang. This simply give more space for it to end its cessation and continue with moving inside a bigger space.

This eliminates the set paradoxes which cannot be resolved otherwise. This now seems to say that, fundamentally, until this symmetry breaking occurs, we cannot have access to better foundations for mathematics. The only thing necessary for categories to replace sets is to add the nothing category to the collection of all categories, which (3) expresses.

This immediately reminded me of the blue-eyed islanders problem, where one new bit of information is introduced by the person who speaks first. That person only states the obvious, but that is a new bit. The bit says that the game finished. This can simply be stated as:

The meaning of no category is every category. (4)

Which is just the 'nothing' category added to every other category, similar to the process of set expansion based on introducing the empty set. This restructuring replaces the set expansion with category expansion (in a much bigger logical space).

The parallel with the blue-eyed islanders made me write this note in the remote case it is useful. Please excuse any inaccuracies. I am not a professional mathematician.

Thanks for reading.

[1] https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-isla...