Lectures on the Hyperreals

Goldblatt, Robert

doi:10.1007/978-1-4612-0615-6

Cited by 210 publications

(117 citation statements)

References 0 publications

Supporting

Mentioning

117

Contrasting

Order By: Relevance

“…For the most part, we employ that used in [9]. To obtain the hyperreals, 2 all sequences of real numbers are partitioned into equivalence classes with respect to a chosen and fixed 'nonprincipal 3 ultrafilter' F on the set N of natural numbers: N = {0, 1, 2, .…”

Section: Some Elements Of Nonstandard Analysis and Other Preliminariesmentioning

confidence: 99%

“…To recover the connections to r ω , we may use shorts instead of 8 In graph-theoretic terms, this is like an 'elementary contraction" of the shorted branch except that restored self-loops and parallel branches are not removed. 9 A 0-tip is an equivalence class of one-ended paths, where two one-ended paths are taken to be equivalent if they are identical except for finitely many nodes and branches. Tips of higher ranks are defined similarly.…”

Section: Restorations Of Two Ladder Networkmentioning

confidence: 99%

“…Other expositions are based more upon mathematical analysis. See, for example, [9] and [10]. Shorter introductions can be found in [12] and [13].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hyperreal Operating Points in Transfinite Resistive Networks

Zemanian

2003

Circuits Syst Signal Process

View full text Add to dashboard Cite

Transfinite resistive electrical networks may (or may not) have operating points, and, even when voltages and currents do exist within them, Kirchoff's laws may not be satisfied everywhere. Moreover, rather severe restrictions have to be placed on those networks in order to obtain such results. However, by borrowing some techniques of nonstandard analysis, we can replace the real-valued voltages and currents by hyperreal-valued voltages and currents. As a result, Kirchhoff's laws are always satisfied so long as the transfinite network is "restorable" in the sense that the transfinite network is the end result of an expanding sequence of finite networks. No other restrictions on the transfinite network are needed. Several transfinite networks are then examined under nonstandard analysis, and explicit hyperreal currents and voltages are established for them.

show abstract

Section: Some Elements Of Nonstandard Analysis and Other Preliminariesmentioning

confidence: 99%

Section: Restorations Of Two Ladder Networkmentioning

confidence: 99%

See 1 more Smart Citation

Hyperreal Operating Points in Transfinite Resistive Networks

Zemanian

2003

Circuits Syst Signal Process

View full text Add to dashboard Cite

show abstract

“…On the other hand, internal sets are the members of the enlargements of standard structures, where the enlargements are general models (Henkin models) of type theory or higher order logic. So non-standard analysis is based on higher order logic while the basic algebraization occurring in the theory is the Boolean algebra (see [10], [15]). …”

Section: Introductionmentioning

confidence: 99%

Non-standard Stochastics with a First Order Algebraization

Ferenczi

2010

Stud Logica

View full text Add to dashboard Cite

Internal sets and the Boolean algebras of the collection of the internal sets are of central importance in non-standard analysis. Boolean algebras are the algebraization of propositional logic while the logic applied in non-standard analysis (in non-standard stochastics) is the first order or the higher order logic (type theory). We present here a first order logic algebraization for the collection of internal sets rather than the Boolean one. Further, we define an unusual probability on this algebraization.

show abstract

“…We aim to demonstrate that various differential geometric concepts and various proofs become simpler and more transparent in this framework. For good introductions to nonstandard analysis see eg Albeverio et al [1], Goldblatt [4], Gordon, Kusraev and Kutateladze [5], Keisler [7], Loeb and Wolff [9], Väth [16]. For an advanced study of axiomatic treatments see Kanovei and Reeken [6].…”

Section: Introductionmentioning

confidence: 99%

Differential geometry via infinitesimal displacements

Nowik

Katz

2015

JLA

View full text Add to dashboard Cite

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, eg 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. 26E35, 58A05 (primary) Mathematics Subject Classification

show abstract

Lectures on the Hyperreals

Cited by 210 publications

References 0 publications

Hyperreal Operating Points in Transfinite Resistive Networks

Hyperreal Operating Points in Transfinite Resistive Networks

Non-standard Stochastics with a First Order Algebraization

Differential geometry via infinitesimal displacements

Contact Info

Product

Resources

About