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The subset of numbers for which you can compute successively better approximations, such as pi and e, is countable. It’s called the set of computable numbers an

The subset of numbers for which you can compute successively better approximations, such as pi and e, is countable. It’s called the set of computable numbers and contains everything that we could even in theory ever describe or “use” somehow. The actually uncountable subset of reals is by definition forever out of our reach, which is what makes uncountable sets… rather ontologically suspicious.

adrian_b 5y ago
This is a matter of opinion. I do not find anything ontologically suspicious about uncountable sets. What I would really found ontologically suspicious would be the existence of convergent sequences of numbers without a limit or the inexistence of continuity, e.g. the impossibility of knowing whether moving from one side of a curve to the other side of the curve will intersect the curve or not. Even if most programmers are among the people who use the least the mathematics based on real numbers, for most of the mankind the mathematical models that would not be possible without the concepts of limit and continuity have become essential for survival, regardless if they are able to understand them or not. I strongly dislike the meaningless term "real numbers", which was created as an opposite to the equally meaningless term "imaginary numbers". However what matters are not the numbers or whether they can be counted or not. What matters are the concepts of limits and continuity. Already Aristotle classified the quantities into discrete quantities and continuous quantities, using a definition of continuity not too different from the modern, even if it was non-rigorous. In Antiquity, what are called now "real numbers" were called "measures", i.e. results of measurement operations, in contrast with the "numbers" obtained by counting.
- smilliken 5y ago
  Limits work just fine without dealing with infinity. When you do an epsilon delta proof, you only ever deal in the finite. You can also have continuity while only dealing in finite constructions. The real numbers are a very convenient abstraction, but they are also a leaky abstraction (edit: when paired with axiom of choice), which is anathema to mathematics. We can still study the real numbers, they are still valid in some abstract sense, and they are convenient, but an element of the set of reals minus computables can never be constructed, so you'll never actually need to deal with them.

adrian_b 5y ago

This is a matter of opinion. I do not find anything ontologically suspicious about uncountable sets. What I would really found ontologically suspicious would be the existence of convergent sequences of numbers without a limit or the inexistence of continuity, e.g. the impossibility of knowing whether moving from one side of a curve to the other side of the curve will intersect the curve or not. Even if most programmers are among the people who use the least the mathematics based on real numbers, for most of the mankind the mathematical models that would not be possible without the concepts of limit and continuity have become essential for survival, regardless if they are able to understand them or not. I strongly dislike the meaningless term "real numbers", which was created as an opposite to the equally meaningless term "imaginary numbers". However what matters are not the numbers or whether they can be counted or not. What matters are the concepts of limits and continuity. Already Aristotle classified the quantities into discrete quantities and continuous quantities, using a definition of continuity not too different from the modern, even if it was non-rigorous. In Antiquity, what are called now "real numbers" were called "measures", i.e. results of measurement operations, in contrast with the "numbers" obtained by counting.

smilliken 5y ago
Limits work just fine without dealing with infinity. When you do an epsilon delta proof, you only ever deal in the finite. You can also have continuity while only dealing in finite constructions. The real numbers are a very convenient abstraction, but they are also a leaky abstraction (edit: when paired with axiom of choice), which is anathema to mathematics. We can still study the real numbers, they are still valid in some abstract sense, and they are convenient, but an element of the set of reals minus computables can never be constructed, so you'll never actually need to deal with them.