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Title page 1

Contents 1

ABSTRACT 1

1. INTRODUCTION 2

1.1. COSMOLOGICAL MODELS FOR THE ENTROPIC ARROW OF TIME 2

1.2. THE TROUBLE WITH NON-NORMALIZABLE MEASURES 5

2. TWO INFINITE FAIR LOTTERIES PARADOX IN STANDARD PROBABILITY THEORY 6

2.1. CONGLOMERABILITY IN KOLMOGOROV'S PROBABILITY THEORY 6

2.1.1. Absolute probability 6

2.1.2. Conditional probability 7

2.1.3. Countable conglomerability 8

2.2. TWO INFINITE FAIR LOTTERIES PARADOX 9

3. THREE ALTERNATIVE FORMALISMS THAT DIFFUSE THE PARADOX 11

3.1. DE FINETTI'S MERELY FINITELY ADDITIVE PROBABILITY THEORY 11

3.2. BENCI ET AL.'S NON-ARCHIMEDEAN PROBABILITY THEORY 12

3.2.1. NAP functions are not countably conglomerable 12

3.2.2. NAP functions are conglomerable on Λ-compatible partitions 13

3.2.3. NAP functions are conglomerable on suitably coarse-grained Λ-compatible partitions 14

3.3. NON-NORMALIZABLE QUASI-PROBABILITY 15

4. THE PARADOX IN COSMOLOGY 16

4.1. GOLDSTEIN ET AL.'S CUT-OFF PARADOX 16

4.1.1. Analysis with NAP theory 19

4.2. ACCOUNTING FOR DYNAMICAL SIMILARITY TO RESTORE NORMALIZABILITY? 20

5. CONCLUSIONS 22

ACKNOWLEDGEMENTS 23

REFERENCES 23

Figures 10

Figure 1. Part of the sample space N × N with some events and members of partitions relevant for the two infinite fair lotteries paradox.... 10

Figure 2. Bottom part of the sample space {(𝑥, 𝑦) / 𝑦 ≥ 𝑥²} with some events and members of partitions relevant for the cut-off paradox... 18

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