figshare
Browse
1/1
2 files

Construction of the Transreal Numbers from Rational Numbers via Dedekind Cuts

Version 2 2020-06-18, 12:43
Version 1 2020-04-16, 14:55
journal contribution
posted on 2020-06-18, 12:43 authored by James AndersonJames Anderson
PURPOSE

Draft journal article, dated 18 June 2020, for further comment and collaboration. Email the authors using the hyperlinks in the PDF.



ABSTRACT

The first constructive definition of the real numbers was in terms of Dedekind cuts. A Dedekind cut is an ordered partition of the rational numbers into two non-empty sets, the lower set and the upper set. However, outlawing empty sets makes the definition partial.

We totalise the set of ordered partitions by admitting two cuts: the negative infinity cut is the cut with an empty lower set and a full upper set; the positive infinity cut is the cut with a full lower set and an empty upper set. These correspond to the affine infinities of the extended-real numbers. We further admit the nullity cut that has both an empty lower set and an empty upper set. We say that the set of all Trans-Dedekind cuts comprises the set of all Dedekind cuts, together with the three strictly Trans-Dedekind cuts: positive infinity, negative infinity, and nullity.

The arithmetical operations and order relation on Dedekind cuts are usually defined only on the lower or else upper sets, which is incoherent when applied to strictly Trans-Dedekind cuts. We totalise these operations and relation over lower and upper sets. We call our totalised Dedekind arithmetic, Trans-Dedekind arithmetic.

We find that the Trans-Dedekind arithmetic of Trans-Dedekind cuts is isomorphic to transreal arithmetic, which is total. This construction gives transreal arithmetic the same ontological status as real arithmetic.


History

Usage metrics

    Licence

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC