# Black Holes and Complexity via Constructible Universe

^{*}

^{†}

## Abstract

**:**

## 1. Introduction and Motivations

## 2. Mathematical Formalism of QM in the Minimal Constructive Model of ZFC

- (a)
- Outside the model ${L}_{\alpha}$, there exists a Hilbert space H and an isometric monomorphism U from ${H}_{{L}_{\alpha}}$ into H.
- (b)
- Outside the model ${L}_{\alpha}$, there exists an isometric monomorphism V from $B\left({H}_{{L}_{\alpha}}\right)$ into $B\left(H\right)$.
- (c)
- Let S and Q be collections of state preparation procedures and question measuring procedures, respectively. There exist maps $f:S\to B\left(H\right)$ and $g:Q\to B\left(H\right)$ such that, for any $s\in S$ and $q\in Q$, the values $f\left(s\right)$ and $g\left(q\right)$ are respectively the density and projection operators (outside ${L}_{\alpha}$). There also exist analogical maps ${f}_{{L}_{\alpha}}$ and ${g}_{{L}_{\alpha}}$ such that appropriate values are density and projection operators from $B\left({H}_{{L}_{\alpha}}\right)$ (inside the model ${L}_{\alpha}$). Moreover, in the respective domains of ${f}_{{L}_{\alpha}}$ and ${g}_{{L}_{\alpha}}$, the mean values of these projections in the set of equally prepared states by these density operators are equal outside ${L}_{\alpha}$, i.e.,$${\mathrm{Tr}}_{{L}_{\alpha}}\left({f}_{{L}_{\alpha}}\left(s\right){g}_{{L}_{\alpha}}\left(q\right)\right)=\mathrm{Tr}\left(f\left(s\right)g\left(q\right)\right).$$

## 3. Models of ZFC and Computational Complexity of BH’s State Space

**Corollary**

**1**

#### 3.1. BH Horizon

#### 3.2. Computational Complexity and BHs

[RAND] A binary infinite sequence $r\in {R}_{M\left[r\right]}$ is random with respect to the model M if r omits all Borel sets of measure zero coded in M.

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BH | Black Hole |

c.e. | computably enumerable |

CTM | Countable Transitive Model |

GR | General Relativity |

MASA | Maximal Abelian SubAlgebra |

s.a. | self-adjoint operator |

QM | Quantum Mechanics |

ZFC | Zermelo–Fraenkel set theory with the axiom of Choice |

## Appendix A

**Lemma**

**A1.**

## Appendix B

**Lemma**

**A2.**

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Król, J.; Klimasara, P.
Black Holes and Complexity via Constructible Universe. *Universe* **2020**, *6*, 198.
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**AMA Style**

Król J, Klimasara P.
Black Holes and Complexity via Constructible Universe. *Universe*. 2020; 6(11):198.
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**Chicago/Turabian Style**

Król, Jerzy, and Paweł Klimasara.
2020. "Black Holes and Complexity via Constructible Universe" *Universe* 6, no. 11: 198.
https://doi.org/10.3390/universe6110198