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Problems with infinity where inductive expansion or contraction are concerned are discussed. Three special cases when thinking about infinity is required, are considered: problems where the task is to estimate the value of a real valued parameter believed to take values in some finite interval; problems where it must be determined which of a countable set of alternative hypotheses is true; and problems where the value of a real valued parameter on the real line must be estimated. In relation to the first, a condition of finite discriminability is introduced and made use of. In relation to the second, a rejection rule is formulated which requires that conditional probabilities on finite subsets of the ultimate partition be well defined, even though the unconditional probabilities for the finite subset are equal to zero; this rule is defended. The techniques involved in formulating this rule are employed in solving problems of the third type.

*Keywords: *
conditional probabilities;
finite discriminability;
finite intervals;
infinity;
ultimate partition

*Chapter.*
*5403 words.*

*Subjects: *
Metaphysics

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