Probability Axioms and Set Theory Paradoxes
Abstract
:1. A Puzzle
2. Introduction
2.1. Axioms and Mathematical Intuition
2.2. Mathematical Background
2.3. Freiling’s Argument for
- (i)
- For all ,
- (ii)
- (i)
- For all ,
- (ii)
2.4. Revisiting Kolmogorov’s Axioms
3. Minimalist Probability and New Axioms
3.1. Uniform Probability on
- (i)
- If is finite, and , then
- (ii)
- If and , then .
3.2. Two Axioms
- (i)
- there exists an such that for all , where ⊕ denotes the bitwise XOR operation, or
- (ii)
- there exists a permutation such that for all .
4. Results
4.1. A Vitali-Type Paradox
- For every , there exists a (unique) ,
- such that for all , .
4.2. The Banach–Tarski Paradox in
- (i)
- ,
- (ii)
- for all and
- (iii)
- for all
- (i)
- , , ,
- (ii)
- , , ,
- (iii)
- , , ,
5. Conclusions
- (i)
- reject the Infinity or Powerset axioms, so cannot be constructed,
- (ii)
- reject Axiom 4, or
- (iii)
- reject Axioms 6 and 7.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Herman, A.; Caughman, J. Probability Axioms and Set Theory Paradoxes. Symmetry 2021, 13, 179. https://doi.org/10.3390/sym13020179
Herman A, Caughman J. Probability Axioms and Set Theory Paradoxes. Symmetry. 2021; 13(2):179. https://doi.org/10.3390/sym13020179
Chicago/Turabian StyleHerman, Ari, and John Caughman. 2021. "Probability Axioms and Set Theory Paradoxes" Symmetry 13, no. 2: 179. https://doi.org/10.3390/sym13020179
APA StyleHerman, A., & Caughman, J. (2021). Probability Axioms and Set Theory Paradoxes. Symmetry, 13(2), 179. https://doi.org/10.3390/sym13020179