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Hyperreals Great and Small

  • Robert Goldblatt
Part of the Graduate Texts in Mathematics book series (GTM, volume 188)

Abstract

Members of *ℝ are called hyperreal numbers, while members of ℝ are real and sometimes called standard. *ℚ consists of hyperrationals, *ℤ of hyperintegers, and *ℕ of hypernaturals. That *ℚ consists precisely of quotients m/n of hyperintegers m, n ∈ *ℤ follows by transfer of the sentence
$$ \forall x \in \mathbb{R}\left[ {x \in \mathbb{Q} \leftrightarrow \exists y,z \in \mathbb{Z}\left( {z \ne 0 \wedge x = y/z} \right)} \right]. $$
It is now time to examine the basic arithmetical and algebraic structure of *ℝ, particularly in its relation to the structure of ℝ.

Keywords

Cauchy Sequence Quotient Ring Positive Real Nonstandard Analysis Real Comparison 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Robert Goldblatt
    • 1
  1. 1.School of Mathematical and Computing SciencesVictoria UniversityWellingtonNew Zealand

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