<2>
We know that a brain in a vat can experience a 3D world "out there." But where *exactly* does this perceived, 3D world "exist"? (Does it exist somewhere "inside" a four-dimensional framework?)

<3>
As we know, the "spacetime" model (i.e., representation) of reality is very useful. However, when discussing consciousness and the strange "world" of mental phenomena, it might be more useful to consider alternative representations of reality.

<4>
How do the quantum observations of two people relate to one another?

<5>
Let us consider two individuals, Ann and Betty. In the spacetime model of reality, Ann's observations relate to Betty's observations in a particular way: Their observations are defined as "3D slices" of a common, 4D space (i.e., spacetime).(1)

<6>
In this article, I consider an alternative model of reality: The set-theoretical (ST) representation.(2) In the ST model, observations do not exist within a common physical space.

<7>
Let us consider our observer Ann. During the course of her lifetime ("L(A)"), Ann perceives a succession of 3D images.

<8>
However, according to quantum theory, there are many alternative sequences of 3D images that she can observe over the period L(A).

<9>
Let the symbols a1, a2, a3, etc. represent these various sequences of 3D images. Together, a1, a2, a3, etc. make up a set that we can call "Set-A."

<10>
Thus: Set-A = {a1, a2, a3, etc.}.

<11>
Quantum theory permits Ann to observe any one of these sequences over the period L(A).(3)

<12>
Betty, our second observer, has a lifetime L(B). Betty observes a succession of 3D images over the duration L(B).

<13>
However, according to quantum theory, there are many alternative sequences of 3D images that she can observe over the period L(B).

<14>
Let the symbols b1, b2, b3, etc. represent these various sequences of 3D images. Together, b1, b2, b3, etc. make up a set that we can call "Set-B."

<15>
Thus: Set-B = {b1, b2, b3, etc.}.

<16>
Quantum theory permits Betty to observe any one of these sequences over the period L(B).

<17>
Suppose Ann observes the sequence a2 over the duration L(A); and suppose Betty observes the sequence b3 over L(B).

<18>
Then we can define Set-O as follows: Set-O = {a2, b3}. This is the set of "sequences of 3D images" that are actually observed. Set-O consists of the elements a2 and b3.

<19>
The set-theoretical model described above can easily be extended to represent any number of observers. For example, we can add two more observers, Chris and David. So we now have a reality consisting of four observers: Ann, Betty, Chris, and David.

<20>
L(A) represents Ann's lifetime. L(B) represents Betty's lifetime. L(C) represents Chris's lifetime. And L(D) represents David's lifetime.

<21>
In this scenario, we have a total of five sets. Here are the first four:

<22>
(Note: Ann can observe any one of the sequences in Set-A over the duration L(A). Betty can observe any one of the sequences in Set-B over the duration L(B). Chris can observe any one of the sequences in Set-C over the duration L(C). And David can observe any one of the sequences in Set-D over the duration L(D).)

<23>
Let us suppose the following: Ann observes a2 over the duration L(A); Betty observes b3 over L(B); Chris observes c1 over L(C); and David observes d3 over L(D).

<24>
Then we can define the fifth set (Set-O) as follows:

<25>
Set-O = {a2, b3, c1, d3}. This is the set of "sequences of 3D images" that are actually observed. This set represents reality as it is experienced by the above four individuals over the course of their lives.

<26>
We can combine elements from the first four sets in many different ways to create the fifth set, Set-O. The set of all such combinations constitutes a class, Class-T.

<27>
The elements (i.e., quadruplets) in Class-T that are permitted by the laws of quantum theory constitute a class, Class-P. Class-P is a subclass of Class-T.(4)

<28>
Let us suppose the following:

<29>
This scenario (S1) seems logically possible in the set-theoretical model.(3) However, S1 does not seem possible in the spacetime model.(4)