The Transfer Principle holds for definable nonstandard models under Countable Choice

Herzberg F. The Transfer Principle holds for definable nonstandard models under Countable Choice. Center for Mathematical Economics Working Papers. Vol 560. Bielefeld: Center for Mathematical Economics; 2016.Łos’s theorem for (bounded) D-ultrapowers, D being the ultrafilter introduced by Kanovei and Shelah [Journal of Symbolic Logic, 69(1):159–164, 2004], can be established within Zermelo–Fraenkel set theory plus Countable Choice ($ZF+AC_\omega$). Thus, the Transfer Principle for both Kanovei and Shelah’s definable nonstandard model of the reals and Herzberg’s definable nonstandard enlargement of the superstructure over the reals [Mathematical Logic Quarterly, 54(2):167–175; 54(6):666– 667, 2008] can be shown in $ZF+AC_\omega$. This establishes a conjecture by Mikhail Katz [personal communication]

When is .999... less than 1?

We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's "semicolon" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999 . . . be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student.Comment: 28 page

Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.Comment: 35 pages, 2 figures, to appear in Notices of the American Mathematical Society 61 (2014), no.

Strict Dominance and Symmetry

The Strict Dominance Principle that a wager always paying better than another is rationally preferable is one of the least contro- versial principles in decision theory. I shall show that (given the Axiom of Choice) there is a contradiction between Strict Dominance and plau- sible isomorphism or symmetry conditions, by showing how in several natural cases one can construct isomorphic wagers one of which strictly dominates the other. In particular, I will show that there is a pair of wagers on the outcomes of a uniform spinner which differ simply in where the zero degrees point of the spinner is defined to be but where one wager dominates the other. I shall also argue that someone who ac- cepts Williamson’s famous argument that the probability of an infinite sequence of heads is zero should accept the symmetry conditions, and thus has reason to weaken the Strict Dominance Principle, and I shall propose a restriction of the Principle to “implementable” wagers. Our main result also has implications for social choice principles

Strict Dominance and Symmetry

The Strict Dominance Principle that a wager always paying better than another is rationally preferable is one of the least contro- versial principles in decision theory. I shall show that (given the Axiom of Choice) there is a contradiction between Strict Dominance and plau- sible isomorphism or symmetry conditions, by showing how in several natural cases one can construct isomorphic wagers one of which strictly dominates the other. In particular, I will show that there is a pair of wagers on the outcomes of a uniform spinner which differ simply in where the zero degrees point of the spinner is defined to be but where one wager dominates the other. I shall also argue that someone who ac- cepts Williamson’s famous argument that the probability of an infinite sequence of heads is zero should accept the symmetry conditions, and thus has reason to weaken the Strict Dominance Principle, and I shall propose a restriction of the Principle to “implementable” wagers. Our main result also has implications for social choice principles

Internal laws of probability, generalized likelihoods and Lewis' infinitesimal chances-A response to Adam Elga

Herzberg F. Internal laws of probability, generalized likelihoods and Lewis' infinitesimal chances-A response to Adam Elga. The British Journal for the Philosophy of Science. 2007;58(1):25-43