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4. Or also a positive real number is an equivalence class of aggregates which have finite value. Where an aggregate a has finite value if there exists an aggregate [math] b [/math], containing finitely many elements, such that [math] a \leq b [/math]. Where aggregates are collections of integer reciprocals (that are not evaluated).

Tweddle, J. C. (2011). Weierstrass’s construction of the irrational numbers. Mathematische Semesterberichte, 58(1), 47-58.

5. The abstract algebraic synthetic approach as a Dedekind-complete compatibly linearly ordered field (not exactly the same as dedekind cuts)

Hall, J. F. (2011). Completeness of ordered fields. arXiv preprint arXiv:1101.5652.

Tarski's axiomatization

Wikipedia contributors. (2018, November 4). Construction of the real numbers. In Wikipedia, The Free Encyclopedia. Retrieved 10:08, November 7, 2018, from https://en.wikipedia.org/w/index.php?title=Construction_of_the_real_numbers&oldid=867304826

Leibniz completeness, Hilbert completeness, Bolzano completeness, Heine-Borel completeness, Cantor completnesses,

Hall, J. F., & Todorov, T. D. (2011). Completeness of the Leibniz field and rigorousness of infinitesimal calculus. arXiv preprint arXiv:1109.2098.

Bachmann's construction by refinement of rational nests, Bourbaki's approach, Maier-Maier's variation of Dedekind cuts, Shiu's construction by infinite series, Flatin-Metropolis-Ross-Rota's wreath construction, De Brujin's construction by additive expansions, Rieger's construction by continued fractions, Schaunuel et al's construction using approximate endomorphisms of [math] \mathbb{Z} [/math], Knopfmacher-Knopfmacher's various constructions, Pintilie's construction by infinite series, Arthan's irrational construction, Conway's Surreal numbers

Weiss, I. (2015). Survey Article: The real numbers–A survey of constructions. Rocky Mountain Journal of Mathematics, 45(3), 737-762.