
Ask HN: Relationship between set theory and category theory - fmihaila
I have an idea about the relationship between set theory and category theory and I would like some feedback. I would like others to see it too, and I don&#x27;t know how to do it. I think it&#x27;s at least interesting to look at as a slightly crazy collage, but I was a bit more excited than normal when the idea hit, so I just had to dump it all at once in this image: https:&#x2F;&#x2F;twitter.com&#x2F;FamilialRhino&#x2F;status&#x2F;1101777965724168193 (You will have to zoom the picture in order to be able to read the scribbles.)<p>It has to do with resonance in the energy flowing in emergent networks. Can&#x27;t quite put my finger on it, so I&#x27;ll be here to answer any questions.<p>Thanks for reading.
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westurner
"Categorical set theory" > "References"
[https://en.wikipedia.org/wiki/Categorical_set_theory#Referen...](https://en.wikipedia.org/wiki/Categorical_set_theory#References)

From "Homotopy category" > "Concrete categories"
[https://en.wikipedia.org/wiki/Homotopy_category#Concrete_cat...](https://en.wikipedia.org/wiki/Homotopy_category#Concrete_categories)
:

> _While the objects of a homotopy category are sets (with additional
> structure), the morphisms are not actual functions between them, but rather
> a classes of functions (in the naive homotopy category) or "zigzags" of
> functions (in the homotopy category). Indeed, Freyd showed that neither the
> naive homotopy category of pointed spaces nor the homotopy category of
> pointed spaces is a concrete category. That is, there is no faithful functor
> from these categories to the category of sets._

~~~
fmihaila
My "understanding" of category theory is extremely shallow, but that's exactly
why I think my proposal makes sense. It is the kind of thing that everybody
ignores for decades precisely because it's transparently obvious, like a fish
that doesn't understand water.

Here is the statement:

The meaning of no category is every category.

reference: [https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-
isla...](https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-
puzzle/)

This was already understood by everybody in the field, no doubt. It's just
that somebody has to actually _say it_ to someone else in order for the
symmetry to break. The link above has the exact description of this, from
Terence Tao.

------
fmihaila
I also asked this question on MathOverflow, in case anyone is interested to
follow.

[https://mathoverflow.net/questions/324506/is-this-lifting-
of...](https://mathoverflow.net/questions/324506/is-this-lifting-of-set-into-
cat-a-valid-construction).

------
fmihaila
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...](https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-
puzzle/)

