Reals-as-Oracles Alternatives

Reals-as-Oracles reviews and mentions

• What are the real numbers, really? (2024)
  1 project | news.ycombinator.com | 14 Aug 2025

  I came up with a different definition that is a kind of inverse of Dedekind cuts. It is the idea that a real number is the set of all rational intervals that contain it. Since this is circular, there are properties that I came up with which say when a set of rational intervals qualifies to be called a real number in my setup. I have an unreviewed paper which creates a version that is a bridge between numerical analysis and the theoretical definition of a real number. Another unreviewed paper shows the equivalence between my definition and Dedekind cuts. You can read both at [1].

  There is a long tradition of using intervals for dealing with real numbers. It is often used by constructivists and can be thought of viewing a real number as a measurement.

  1: https://github.com/jostylr/Reals-as-Oracles

• Mathematician solves algebra's oldest problem using intriguing number sequences
  1 project | news.ycombinator.com | 3 May 2025
• "A calculator app? Anyone could make that."
  9 projects | news.ycombinator.com | 16 Feb 2025

  I have been working on a new definition of real numbers which I think is a better foundation for real numbers and seems to be a theoretical version of what you are doing practically. I am currently calling them rational betweenness relations. Namely, it is the set of all rational intervals that contain the real number. Since this is circular, it is really about properties that a family of intervals must satisfy. Since real numbers are messy, this idealized form is supplemented with a fuzzy procedure for figuring out whether an interval contains the number or not. The work is hosted at (https://github.com/jostylr/Reals-as-Oracles) with the first paper in the readme being the most recent version of this idea.

  The older and longer paper of Defining Real Numbers as Oracles contains some exploration of these ideas in terms of continued fractions. In section 6, I explore the use of mediants to compute continued fractions, as inspired by the old paper Continued Fractions without Tears ( https://www.jstor.org/stable/2689627 ). I also explore a bit of Bill Gosper's arithmetic in Section 7.9.2. In there, I square the square root of 2 and the procedure, as far as I can tell, never settles down to give a result as you seem to indicate in another comment.

  For fun, I am hoping to implement a version of some of these ideas in Julia at some point. I am glad to see a version in Python and I will no doubt draw inspiration from it and look forward to using it as a check on my work.

• "A Course of Pure Mathematics" – G. H. Hardy (1921) [pdf]
  3 projects | news.ycombinator.com | 31 Dec 2024
• Math Limitations
  1 project | news.ycombinator.com | 22 Jul 2023

  I think having a language that helps understand those limitations is a useful achievement. Much of mathematics does have that. A notable exception is the definition of real numbers. They are usually presented as a string of infinite decimals, or a converging sequence, or a set of numbers less than something. All of those notions obscure the basic limitation of knowing the real number and give a veneer of similarity to rational number. Rational numbers are numbers that we can have in our hand while irrational numbers are ones which we can never have. It is important to have a setup that respects that difference.

  This is what motivated me to come up with a new definition of real numbers, namely, they are objects (I call them oracles) that answer Yes or No when asked if the number ought to be between two given rational numbers. Abstracting out what properties such an object should have, one can come up with a space of these oracles, define an arithmetic, and prove that they satisfy the axioms of real numbers.

  For details: https://github.com/jostylr/Reals-as-Oracles/

  In many ways, this is giving a definitional support to the use of interval analysis which is, of course, a very practical concern. It also brings our some cool stuff about mediants and continued fractions (nothing new about that, but nicely motivated).

  It also fits in with the adjacent post about busy beaver numbers and its conclusion about knowing a number is in an interval.

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