Homotopy Type Theory real numbers > history (Rev #7, changes)

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Definition

The type of real numbers $\mathbb{R}$ is defined a as the terminallocally (-1)-connected? Hausdorff sober? Archimedean ordered field or with the a terminalArchimedean ordered integral domaincompact? real unit interval? $[0, 1]$.

Other types called ‘real numbers’

There are many other different types which are called real numbers in the literature, many of which are do not satisfy the same properties as listed above for the real numbers numbers. defined above. These include:

• Cauchy real numbers (disambiguation)

• Dedekind real numbers (disambiguation page)

• Eudoxus real numbers

• localic real numbers? (forms (opens are binary relations, type is aframe and lies in a higher universe in the hierarchy) and sigma-localic real numbers? (forms (opens a are functions intoSierpinski space, type is a $\sigma$-frame and lies in the same universe)

• MacNeille real numbers? or Dedekind-MacNeille real numbers

• real unit interval? based real numbers

  • Euclidean real numbers? or Escardo-Simpson real numbers

  • The various types of real numbers defined by Peter Freyd using various definitions of the co-algebraic real unit interval.

References