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]

Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem

By using nonstandard analysis, and in particular iterated hyperextensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of our technique, we give a (rather) short proof of Milliken-Taylor’s Theorem and a ultrafilter version of Rado’s Theorem about partition regularity of diophantine equations

Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

We examine some of Connes' criticisms of Robinson's infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes' own earlier work in functional analysis. Connes described the hyperreals as both a "virtual theory" and a "chimera", yet acknowledged that his argument relies on the transfer principle. We analyze Connes' "dart-throwing" thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being "virtual" if it is not definable in a suitable model of ZFC. If so, Connes' claim that a theory of the hyperreals is "virtual" is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren't definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in his Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes' criticism of virtuality. We analyze the philosophical underpinnings of Connes' argument based on Goedel's incompleteness theorem, and detect an apparent circularity in Connes' logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace (featured on the front cover of Connes' magnum opus) and the Hahn-Banach theorem, in Connes' own framework. We also note an inaccuracy in Machover's critique of infinitesimal-based pedagogy.Comment: 52 pages, 1 figur

A Semi-Constructive Approach to the Hyperreal Line

Using an alternative to Tarskian semantics for first-order logic known as possibility semantics, I introduce an approach to nonstandard analysis that remains within the bounds of semiconstructive mathematics, i.e., does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. I define the Fr´echet hyperreal line †R as a possibility structure and show that it shares many fundamental properties of the classical hyperreal line, such as a Transfer Principle and a Saturation Principle. I discuss the technical advantages of †R over some other alternative approaches to nonstandard analysis and argue that it is well-suited to address some of the philosophical and methodological concerns that have been raised against the application of nonstandard methods to ordinary mathematics

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

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.

Timothy Williamson’s coin-flipping argument: refuted prior to publication

In a well-known paper, Timothy Williamson (Analysis 67:173–180, 2007) claimed to prove with a coin-flipping example that infinitesimal-valued probabilities cannot save the principle of Regularity, because on pain of inconsistency the event ‘all tosses land heads’ must be assigned probability 0, whether the probability function is hyperreal-valued or not. A premise of Williamson’s argument is that two infinitary events in that example must be assigned the same probability because they are isomorphic. It was argued by Howson (Eur J Philos Sci 7:97–100, 2017) that the claim of isomorphism fails, but a more radical objection to Williamson’s argument is that it had been, in effect, refuted long before it was published