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?)

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.

***

How do the quantum observations of two individuals relate to one another?

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

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.

***

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

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

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."

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

Quantum theory permits O_A to observe any one of these sequences over the period L(A).(3)

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

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

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."

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

Quantum theory permits O_B to observe any one of these sequences over the period L(B).

Suppose O_A observes the sequence a2 over the duration L(A); and suppose O_B observes the sequence b3 over L(B).

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.

The set-theoretical model described above can easily be extended to represent any number of observers. For example, we can add two more observers (i.e., two more people), O_C and O_D. So we now have a reality consisting of four observers: O_A, O_B, O_C, and O_D.

L(A) represents O_A's lifetime. L(B) represents O_B's lifetime. L(C) represents O_C's lifetime. And L(D) represents O_D's lifetime.

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

1. Set-A = {a1, a2, a3, etc.}

2. Set-B = {b1, b2, b3, etc.}

3. Set-C = {c1, c2, c3, etc.}

4. Set-D = {d1, d2, d3, etc.}

(Note: O_A can observe any one of the sequences in Set-A over the duration L(A). O_B can observe any one of the sequences in Set-B over the duration L(B). O_C can observe any one of the sequences in Set-C over the duration L(C). And O_D can observe any one of the sequences in Set-D over the duration L(D).)

Let us suppose the following: O_A observes a2 over the duration L(A); O_B observes b3 over L(B); O_C observes c1 over L(C); and O_D observes d3 over L(D).

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

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.

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.

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)

1. The spacetime model makes the following three assumptions: a) Many observers exist; b) each individual makes his or her own observations of reality; c) the observations of several individuals "fit together" to form a "self-consistent network of observations."

2. I do not know if the set-theoretical model described in this article
resembles Professor Hitoshi Kitada's theory in any
way.

(http://groups.yahoo.com/group/time)

3. Quantum theory describes a strange reality of "wave-particles," "uncertainties," and "probabilities of events." "Four-dimensional spacetime" might not "belong" in this world of quantum concepts. We should consider abstract alternatives to the spacetime model of reality.

4. Class-T is the totality of possible quadruplets. Class-P is the class of quadruplets that are permitted by the laws of quantum mechanics.

***

Scenario S1

Let us suppose the following:

- O_A observes a sequence of 3D images [I(A)-1, I(A)-2, I(A)-3].(1)

- She perceives these images over the course of a few moments of subjective/psychological time.(2)

- Each image contains an observable clock.

- The clock in image I(A)-1 indicates 2010 AD.

- The clock in image I(A)-2 indicates 2000 AD.

- The clock in image I(A)-3 indicates 2020 AD.

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

1. Quantum time travel is discussed at the following site:

http://tph.tuwien.ac.at/~svozil/publ/2005-tt.pdf

2. For an analysis of subjective time, please see Section IV of my paper "Temporal Passage":

http://www.kjf.ca/61-TAAND.htm

3. The set-theoretical model places no restrictions on what individuals can perceive.

We can use the laws of quantum mechanics to predict what people actually observe in various circumstances.

4. Logical inconsistencies arise in the spacetime model because observations exist within a common 4D space.

O_A perceives a subjective, virtual world that we can call "V1." O_B perceives a subjective, virtual world that we can call "V2." V1 and V2 resemble each other. But why? Why do these two worlds resemble one another? One explanation is this: The two worlds determine each other. (V1 determines V2, and V2 determines V1.(1))

***

Let us consider the following proposition ("P"):

P: "O_A determines O_B's world (V2), and O_B determines O_A's world (V1). O_A determines V2 by restricting O_B's experiences. O_B determines V1 by restricting O_A's experiences."(2)

I suggest that this proposition is true.

***

Here's how O_A restricts O_B's experiences:

If O_A experiences the sequence a1, then O_B is restricted to SetB_1, a subset of SetB. (O_B is only allowed to experience the sequences in SetB_1.)(3)

If O_A experiences the sequence a2, then O_B is restricted to SetB_2, a subset of SetB. (O_B is only allowed to experience the sequences in SetB_2.)

If O_A experiences the sequence a3, then O_B is restricted to SetB_3, a subset of SetB. (O_B is only allowed to experience the sequences in SetB_3.)

Etc.

Here's how O_B restricts O_A's experiences:

If O_B experiences the sequence b1, then O_A is restricted to SetA_1, a subset of SetA. (O_A is only allowed to experience the sequences in SetA_1.)

If O_B experiences the sequence b2, then O_A is restricted to SetA_2, a subset of SetA. (O_A is only allowed to experience the sequences in SetA_2.)

If O_B experiences the sequence b3, then O_A is restricted to SetA_3, a subset of SetA. (O_A is only allowed to experience the sequences in SetA_3.)

Etc.

***

Let SetT = the following set of ordered pairs: {a1b1, a1b2, ... a2b1, a2b2, ... a3b1, a3b2, ... etc.}.(4) The ordered pairs in this set can be placed in two categories: Category 1 and Category 2. The Category 1 pairs satisfy the "restrictions" listed above. The Category 2 pairs do not satisfy the restrictions.(5)(6)

Let SetP = the ordered pairs in the first category and SetQ = the ordered pairs in the second category.(7)

The set-theoretical model described above can be extended to represent any number of observers. Let us consider a scenario with three observers (i.e., three people), O_A, O_B, and O_C. There are three principal sets in this scenario: SetA, SetB, and SetC.

Here's how the three observers restrict one another:

If O_A experiences the sequence a1, then O_B is restricted to SetB_1 and O_C is restricted to SetC_1.(8)

If O_A experiences the sequence a2, then O_B is restricted to SetB_2 and O_C is restricted to SetC_2.

Etc.

If O_B experiences the sequence b1, then O_A is restricted to SetA_1 and O_C is restricted to SetC_3.

If O_B experiences the sequence b2, then O_A is restricted to SetA_2 and O_C is restricted to SetC_4.

Etc.

If O_C experiences the sequence c1, then O_A is restricted to SetA_4 and O_B is restricted to SetB_4.

If O_C experiences the sequence c2, then O_A is restricted to SetA_5 and O_B is restricted to SetB_5.(9)

Etc.

***

Let us consider the ordered pair a1b1.

Does this pair satisfy the requirement in restriction R? (See below.) In order to answer this question, we need to know whether b1 is a member of SetB_1. If b1 is an element of SetB_1, the answer to the question posed above is "yes." If b1 is not an element of SetB_1, the answer to the question is "no."

Restriction R: "If O_A experiences the sequence a1, O_B is restricted to SetB_1, a subset of SetB. (O_B is only allowed to experience the sequences in SetB_1.)

1. V1 determines V2 in such a way that V2 resembles V1. V2 determines V1 in such a way that V1 resembles V2.

2. This is just another way of saying that V1 and V2 determine each other.

3. The restrictions listed are all "If ... then" statements.

4. SetT combines each element in SetA with every element in SetB.

5. We can refer to the list of "restrictions" as a list of "requirements."

6. The ordered pairs in Category 1 satisfy the requirements. The ordered pairs in Category 2 violate the requirements.

7. SetP and SetQ are subsets of SetT.

8. Note: Subsets of SetA: SetA_1, SetA_2, SetA_3, SetA_4, etc.

of SetB:
SetB_1, SetB_2, SetB_3, SetB_4, etc.

of SetC: SetC_1, SetC_2, SetC_3, SetC_4,
etc.

9. In a restriction, the "If" portion of the "If ... then" statement can refer to more than one person. In the following example, the "If" part of the restriction refers to two people:

Restriction R1: "If O_A experiences the sequence a1 and O_B experiences the sequence b1, then O_C is restricted to SetC_5, a subset of SetC."

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