What qualia are possible?

 I find the question of what possible experiences there are to be fascinating. Let me start listing some qualia: There's the taste of chocolate, the feeling of cold steel, the color red, etc. If this list were to be completed, and every qualia that could be experienced was placed on this list, what would the list look like? Would it be finitely long? Or maybe it would have countably infinite members? Or uncountably infinite members?

 Let's roll up our sleeves and try to discover what this list would look like. 

Imagine you're staring straight forward, and a large television screen takes up your entire vision. Currently the screen is completely white. Then a red circle appears on the left of the screen. Despite this, you stare straight ahead. Let's call this visual quale qᵣₗ. Then, the circle teleports to the middle of the screen. Let's call this new visual quale that you're experiencing qᵣₘ. Then the red circle turns into a purple circle without changing its location. Let's call this visual quale qₚₘ. 

What's interesting is that these qualia have varying degrees of similarity to one another. The taste of chocolate is incomparable to the feeling of cold steel, yet qᵣₗ is closer to qᵣₘ than it is to qₚₘ. Here's an explanation for this phenomenon: the three qualia are made of "building blocks." See, we can view qᵣₗ as composed of the building block "red circle" which is joined to the building block of "at location l₁." The former building block, "red circle," reappears again in the composition of qᵣₘ: "red circle" and "at location l₂."

That's interesting. These qualia are made up of the joining together of more fundamental qualitative building blocks. Let's invent a shorter notation that denotes saying that building blocks b₁ and b₂ are joined together: {b₁, b₂}. So, qᵣₘ is {red circle, at location l₂}.

Now, both qᵣₘ and qₚₘ can be decomposed into {red circle, at location l₂} and {purple circle, at location l₂} respectively. However, "red circle" and "purple circle" also share something in common. "Red circle" seems to be decomposable to the more fundamental building blocks of "red" and "circle." This latter building block is reused in "purple circle": "purple" and "circle." So, qᵣₘ is {{red, circle}, at location l₂} and qₚₘ is {{purple, circle}, at location l₂}.* The takeaway here is that some of the qualitative building blocks can be decomposed into even more fundamental building blocks. We can have quale of the form {{b₁, b₂}, b₃}.

(*Note: One may worry that our vision is "pixelated" in some way. A red circle is really reducible to a bunch of individual "vision pixels" being colored red. The takeaway here still follows, though, since the quale of seeing a circle in that case would be {{red, at location l₁}, {red, at location l₂}, ...}.)

We've already made a ton of progress on our original question. We just need to figure out what the most fundamental qualitative building blocks there are, and then figure out what rules there are for joining them together, and then we can populate our list with every possible combination of the most fundamental qualitative building blocks. 

Now, my pet theory of qualia (which I call Qualia Atomism) is that there is only one fundamental quale, which I will denote ϕ. All other quale are reducible to it. The idea is we can join ϕ to itself, and get {ϕ, ϕ}. Then we can join {ϕ, ϕ} with itself, or ϕ again to either get {{ϕ, ϕ}, {ϕ, ϕ}} or {{ϕ, ϕ}, ϕ}. The rules of how the fundamental quale can be joined to itself are supposed to roughly mirror the axioms of set theory, but instead of an empty set at the bottom of every set, we have ϕ (another difference may be that you can have the same quale in a joining multiple times, so that {ϕ, ϕ, ϕ} is different from {ϕ} in Qualia Atomism, despite that not being true in set theory). So, very quickly we can get some pretty complicated and diverse qualia. The idea is that once you build upwards enough, we get all the qualia we know and love: the smell of cotton candy, the texture of wood, etc.    

However, there's one problem. It seems that the conjunction of theism and Qualia Atomism entails an inconsistency:

1) God can think about any proposition.
2) There are as many possible qualia as there are things which God can think about.
3) There are more propositions than sets.
4) Therefore, there are more possible qualia than there are sets.
5) If Qualia Atomism is true, then there are equally many sets as there are possible qualia.
6) Therefore, Qualia Atomism is false.

The idea behind (2) is that if God is thinking about some proposition, there is something it is like to be thinking about that proposition. The idea behind (5) is that if the rules for joining qualia resemble set theory, then the cardinality of the class of sets is going to be the same as the cardinality of the class of possible qualia.

I will sketch here a way to modify Qualia Atomism in such a way which will hopefully allow us to avoid (5). 

The set theoretic universe could be extended by defining the "∈₂" relation and adding sets₂ into the universe. The idea is that the ∈₂ relation is indistinguishable from the ∈ relation when looking at sets. So, for all x and all y, x ∈₂ y if x ∈ y and both x and y are sets. However, the ∈ is undefined when we add sets₂ to our universe. What are sets₂? Well, for any class of sets c, there is a set₂ containing₂ all sets in c if there is no set which contains all sets in c. We also add set₂-building rules that mirror our ZFC set-building rules and construct a larger hierarchy on top of the existing ZFC hierarchy.

If we think of sets "corresponding" to qualia, like {{Ø}, Ø}} corresponding to {{ϕ}, ϕ} or whatnot (this would need to be tidied up), then it seems conceivable that we could have some qualia corresponding to sets₂. For example, there is no set of all sets. But there is a set₂ of all sets. So, we can have a qualia that is the joining of all qualia that correspond to sets. 

We can also build up sets₂ to sets₃, and then to sets₄ and so on. We can even get transfinite ordinals in here, like adding setsᵪ, which are built up from not just sets, but sets, sets₂, sets₃, sets₄, and so on. Then we can make setsᵪ₊₁. And so on. Our hierarchy can go as high as the ordinals do.

My thinking is that I don't see any Cantorian worries in positing that for any proposition p there will be some a such that a qualia corresponding to a setₐ is what it is like to think about p. I'm not sure if this eliminates Cantorian worries, but it will sure make them really hard to prove. And it really is a God-glorifying metaphysics, because it allows God to have some really complex thoughts!