Hyperreals Great and Small

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


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.


Sine Cose 


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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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