# Set Theory and Many Worlds

## Abstract

**:**

## 1. Many Faces of Many Worlds

## 2. Probability and Uncertainty

#### 2.1. The Logic of Uncertainty

#### 2.2. Many Worlds without Everett

## 3. A Metaphysics for Everettian Fission

- Uncertainty without alternatives:

Uncertainty about the future is the cognitive state of assigning partial degrees of belief to multiple futures; whether those futures are thought of as alternative possibilities or coexistent actualities is an arbitrary choice because the occurrence of a future does not entail that the probability of its occurrence is 1.

- Concrete sets:

Any physical object is a set of Quineian individuals, which is identified with its hierarchy of unit sets. It has all the properties that its elements share, other than those logically excluded, such as the number of elements and value-definiteness.

#### 3.1. From Metaphysics to Physics

#### 3.2. The World as a Wavefunction

## 4. Being Indefinite

#### 4.1. Spin

_{E}be an environmental electron and e

_{e}an elemental electron. Likewise, let p

_{E}be an environmental, spatial point and p

_{e}an elemental point. Every elemental point has an orientation; therefore, for an elemental point oriented parallel to the x-axis, we can write xp

_{e}. Every elemental electron is at an elemental point (e

_{e}@p

_{e}) and is either oriented parallel or orthogonal relative to that point, with parallel being spin-up and orthogonal being spin-down. Therefore, we can write e

_{e}@

_{up}xp

_{e}for an elemental x-spin-up electron and e

_{e}@

_{down}xp

_{e}for an elemental x-spin-down electron. An x-spin-up environmental electron is defined thus:

_{E}(x-spin-up) iff ∀e

_{e}[(e

_{e}∊ e

_{E})&(e

_{e}@xp

_{e})]

`→`[e

_{e}@

_{up}xp

_{e}]

_{e}@

_{up}zp

_{e}} and {e

_{e}@

_{down}zp

_{e}} on {e

_{e}@zp

_{e}} are equal.

_{e}@

_{up}ôp

_{e}} and {e

_{e}@

_{down}ôp

_{e}}, which are non-elemental spin-up and spin-down electrons, since any set of elemental electrons is an electron. They become the post-measurement environmental electrons if a spin measurement is made on the ô orientation. The measures of those subset electrons relative to {e

_{e}@ôp

_{e}} are the probabilities for observing spin-up and spin-down at that orientation.

#### 4.2. Entanglement

#### 4.3. EPR–Bell

_{UP}and Alice

_{DOWN}, whose bodies occupy the local regions A

_{UP}and A

_{DOWN}.. The set of the points that are elements of the points in A is the fusion of the two distinct subsets that are the elements of points in A

_{UP}and A

_{DOWN.}The fissioning of Alice’s body involves the fissioning of spacetime itself. Prior to the measurement, Alice inhabited an environmental region that was a set of elemental regions, each in an elemental universe. Post-measurement Alice

_{UP}’s and Alice

_{DOWN}’s bodies inhabited two distinct environmental regions that contained elemental points in two distinct subsets.

_{UP}and A

_{DOWN}is that they contain two different environmental electrons and different elements of the macroscopic superposition, which is a future temporal counterpart of Alice’s body. A

_{UP}contains all the elements of Alice

_{UP}’s body and none of the elements of Alice

_{DOWN}’s and vice versa. In Bob’s absolute elsewhere, Alice’s body is in a superposition and Alice

_{UP}and Alice

_{DOWN}occupy distinct branches, i.e., distinct subsets of region A.

_{UP}and Bob

_{DOWN}in regions B

_{UP}and B

_{DOWN}. The key point here is that, because of the entanglement, these two subsets of region B differ from A

_{UP}and A

_{DOWN}whilst regions A and B are isomorphic. Necessarily, Alice

_{UP}cannot have measured the same electron as Bob

_{UP}, and Alice

_{DOWN}cannot have measured the same electron as Bob

_{DOWN}. This is a consequence of the two environmental electrons having been in causal contact at their origin.

_{UP}and to Alice

_{DOWN}and Alice’s successor is in superposition relative to B

_{UP}and to B

_{DOWN}. These four observers’ results cannot come into causal contact sooner than half the light-time between regions A and B. To see why, consider Clotilde, halfway along a light path between regions A and B and watching Alice and Bob. When Clotilde sees the results of Alice’s and Bob’s measurements, she fissions into Clotilde

_{AliceUP+BobDOWN}and Clotilde

_{AliceDOWN+BobUP}. As Cai Waegell and Kelvin McQueen put it, “A world containing a Bob and an Alice is only created when the wavefront from Alice’s measurement meets the wavefront from Bob’s measurement” [24] (Section 6). However, it is unclear why they use the term “wavefront”; it is rather a matter of the past lightcones of Alice’s and Bob’s future temporal counterparts coming to overlap.

_{AliceUP}must see Bob

_{DOWN}if Bob measures on the x-axis. However, as we saw in Section 4.1, the structure of the region where Bob’s successor would measure x-spin-down is such that if the measurement had been made on a different axis, the results spin-up and spin-down would have probabilities determined by the subset measures of elements of the electron not measured by Alice

_{UP.}Those elemental electrons would be the ones located at elemental points oriented parallel to the axis chosen by Bob. Therefore, a series of measurements would have to be made on a succession of singlet states for Clotilde’s future temporal counterparts to gather statistical evidence confirming the predicted probabilities.

## 5. Beyond Idealization

## 6. Parting Lines

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Tappenden, P.
Set Theory and Many Worlds. *Quantum Rep.* **2023**, *5*, 237-252.
https://doi.org/10.3390/quantum5010016

**AMA Style**

Tappenden P.
Set Theory and Many Worlds. *Quantum Reports*. 2023; 5(1):237-252.
https://doi.org/10.3390/quantum5010016

**Chicago/Turabian Style**

Tappenden, Paul.
2023. "Set Theory and Many Worlds" *Quantum Reports* 5, no. 1: 237-252.
https://doi.org/10.3390/quantum5010016