234 research outputs found
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
Differential geometry via infinitesimal displacements
We present a new formulation of some basic differential geometric notions on
a smooth manifold M, in the setting of nonstandard analysis. In place of
classical vector fields, for which one needs to construct the tangent bundle of
M, we define a prevector field, which is an internal map from *M to itself,
implementing the intuitive notion of vectors as infinitesimal displacements. We
introduce regularity conditions for prevector fields, defined by finite
differences, thus purely combinatorial conditions involving no analysis. These
conditions replace the more elaborate analytic regularity conditions appearing
in previous similar approaches, e.g. by Stroyan and Luxemburg or Lutz and Goze.
We define the flow of a prevector field by hyperfinite iteration of the given
prevector field, in the spirit of Euler's method. We define the Lie bracket of
two prevector fields by appropriate iteration of their commutator. We study the
properties of flows and Lie brackets, particularly in relation with our
proposed regularity conditions. We present several simple applications to the
classical setting, such as bounds related to the flow of vector fields,
analysis of small oscillations of a pendulum, and an instance of Frobenius'
Theorem regarding the complete integrability of independent vector fields.Comment: Improved presentation in various places. To appear in Journal of
Logic and Analysi
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