Synthese

, Volume 191, Issue 15, pp 3525–3540

# Regular probability comparisons imply the Banach–Tarski Paradox

Article

## Abstract

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: (a) there can be nonmeasurable events that have no probability at all and (b) on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice (AC). We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient to prove the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory.

### Keywords

Set theory Axiom of choice Probability Regularity Bayesianism Incommensurability Credence  Rationality Decision theory Nonmeasurable sets

### References

1. Appiah, K. A. (1985). Assertion and conditionals. Cambridge: Cambridge University Press.
2. Bell, J. L., & Fremlin, D. H. (1972). A geometric form of the axiom of choice. Fundamenta Mathematicae, 77, 167–170.Google Scholar
3. Bernstein, A. R., & Wattenberg, F. (1969). Non-standard measure theory. In W. A. J. Luxemberg (Ed.), Applications of model theory of algebra, analysis, and probability. New York: Holt, Rinehart and Winston.Google Scholar
5. de Finetti, B. (1931). Sul significato soggestivo della probabilità. Fundamenta Mathematicae, 17, 298–329.Google Scholar
6. de Finetti, B. (1975). Theory of Probability. (A. Machi & A. F. M. Smith, Trans.). New York: Wiley.Google Scholar
7. Easwaran, K. (2014). Regularity and hyperreal credences. Philosophical Review, 123, 1–41.
8. Felgner, U., & Truss, J. K. (1999). The independence of the prime ideal theorem from the order-extension principle. Journal of Symbolic Logic, 64, 199–215.
9. Foreman, M., & Wehrung, Friedrich. (1991). The Hahn-Banach Theorem implies the existence of a non-Lebesgue measurable set. Fundamenta Mathematicae, 138, 13–19.Google Scholar
10. Hájek, A. (2003). What conditional probability could not be? Synthese, 137, 273–323.
11. Hájek, A. (2011). Staying regular? Australasian Association of Philosophy Conference, July 2011.Google Scholar
12. Hofweber, T. (MS). Cardinality arguments against regular hyperreal-valued probability measures.Google Scholar
13. Hofweber, T. (2014). Infinitesimal chances. Philosophers’ Imprint, 14, 1–34.Google Scholar
14. Howard, P., & Rubin, J. E. (1998). Consequences of the axiom of choice. Providence, RI: American Mathematical Society.
15. Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford: Oxford University Press.Google Scholar
16. Joyce, J. M. (2010). A defense of imprecise credences in inference and decision. Philosophical Perspectives, 24, 281–323.
17. Kanovei, V., & Shelah, S. (2004). A definable nonstandard model of the reals. Journal of Symbolic Logic, 69, 159–164.
18. Kemeny, J. G. (1955). Fair bets and inductive probabilities. Journal of Symbolic Logic, 20, 263–73.
19. Krauss, P. H. (1968). Representation of conditional probability measures on boolean algebras. Acta Mathematica Academiae Scientiarum Hungarica, 19, 229–241.
20. Laugwitz, D. (1968). Eine nichtarchimedische Erweiterung angeordneter Körper. Mathematische Nachrichten, 37, 225–236.Google Scholar
21. Lewis, D. (1986). On the plurality of worlds. Oxford: Blackwell.Google Scholar
22. Luxemburg, W. (1973). What is nonstandard analysis? American Mathematical Monthly, 80, 38–67.Google Scholar
23. McGee, V. (1994). Learning the impossible. In E. Eells & B. Skyrms (Eds.), Probability and conditionals (pp. 179–199). Cambridge: Cambridge University Press.Google Scholar
24. Parikh, R., & Parnes, M. (1974). Conditional probabilities and uniform sets. In A. Hurd & P. Loeb (Eds.), Victoria symposium on nonstandard analysis (pp. 180–194). Berlin: Springer.
25. Pawlikowski, J. (1991). The Hahn-Banach theorem implies the Banach-Tarski paradox. Fundamenta Mathematicae, 138, 21–22.Google Scholar
26. Pedersen, A. & Paul, M. S. (2013). Strictly coherent preferences, no holds barred. http://www.saet.uiowa.edu/Papers2013/StrictlyCoherentPreferencesNoHoldsBarred_Pedersen.pdf. Accessed 28 April 2014.
27. Pitowsky, I. (1989). Quantum probability., Lecture Notes in Physics Springer: Heidelberg.Google Scholar
28. Pruss, A. R. (2013a). Probability, regularity, and cardinality. Philosophy of Science, 80, 231–240.
29. Pruss, A. R. (2013b). Null probability, dominance and rotation. Analysis, 73, 682–685.
30. Pruss, A. R. (2016). Divine creative freedom. In J. Kvanvig (Ed.), Oxford studies in the philosophy of religion. Oxford: Oxford University Press.Google Scholar
31. Rudin, W. (1987). Real and complex analysis (3rd ed.). New York: McGraw Hill.Google Scholar
32. Shimony, A. (1955). Coherence and the axioms of confirmation. Journal of Symbolic Logic, 20, 1–28.Google Scholar
33. Solovay, R. M. (1970). A model of set-theory in which every set of rals is Lebesgue measurable. Transactions of the American Mathematical Society, 92, 1–56.Google Scholar
34. Skyrms, B. (1980). Causal necessity: A pragmatic investigation of the necessity of laws. New Haven: Yale University Press.Google Scholar
35. Stalnaker, R. C. (1970). Probability and conditionals. Philosophy of Science, 37, 64–80.
36. Suppes, P. (1994). Qualitative theory of subjective probability. In G. Wright & P. Ayton (Eds.), Subjective Probability. Chichester: Wiley.Google Scholar
37. Szpilrajn, E. (1930). Sur l’extension de l’ordre partiel. Fundamenta Mathematicae, 16, 386–389.Google Scholar
38. van Fraassen, B. C. (1976). Representation of conditional probabilities. Journal of Philosophical Logic, 5, 417–430.
39. Villegas, C. (1967). On qualitative probability. American Mathematical Monthly, 74, 661–669.
40. Wagon, S. (1994). The Banach-Tarski paradox. Cambridge: Cambridge University Press.Google Scholar