[HN Gopher] What are the real numbers, really? (2024)
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What are the real numbers, really? (2024)
Author : EthanHeilman
Score : 23 points
Date : 2025-08-14 18:03 UTC (4 hours ago)
(HTM) web link (www.infinitelymore.xyz)
(TXT) w3m dump (www.infinitelymore.xyz)
| thatguysaguy wrote:
| Joel's blog in general is an extremely great read. I highly
| recommend subscribing.
| morpheos137 wrote:
| Real numbers are the concept of quantities built up from
| continuous flows.
| moc_was_wronged wrote:
| Something we made up before we knew Avogadro's Number and no
| longer need.
|
| (That was trolling.)
| glial wrote:
| Hopefully someone better educated than me can answer this -
| several of the definitions in the link feel constructivist, i.e.
| they describe constructions of of real numbers. It seems easy to
| think of methods of constructing non-rational numbers, by e.g.
| using infinite sequences, by taking roots, or whatever.
|
| It seems harder to prove that every real number can be
| constructed via such a method.
|
| Is there a construction-based method that can produce ALL real
| numbers between, say, 0 and 1? This seems unlikely to me, since
| the method of construction would probably be based on some sort
| of enumeration, meaning that you would only end up with countably
| many numbers. But maybe someone else can help me become un-
| confused.
| jtimdwyer wrote:
| I may be misunderstanding your concern, but I believe this is
| what is meant by "Categoricity for the real numbers"
| moc_was_wronged wrote:
| Your original intuition, that only a countable subset of real
| numbers can be described or used in any way, is correct. The
| rest are just "there." They exist, but we can't really use them
| for anything.
|
| It gets weirder. What is a set? For finite sets, we know it
| intuitively. But consider the Axiom of Choice. There is a
| consistent mathematics in which a choice set is a set, and one
| in which the same meta-mathematical object is not a set.
| (Unless, of course, ZF is inconsistent.)
| Kranar wrote:
| The definitions provided appear as though they are
| constructive, but they are not actually constructive, they are
| set-theoretic existence claims that quantify over all
| sequences, in particular over undefinable sets. Specifically,
| the description that appears constructive doesn't actually
| define any particular real number, it only defines the universe
| in which the real numbers live.
|
| Another subtle detail is that while it's true that every real
| number corresponds to (and can be represented by) a Cauchy
| sequence of rationals, the very sequence itself might be
| undefinable.
| jostylr wrote:
| Constructivist basically means being able to be explicit.
| Dedekind cuts and Cauchy sequences are not necessarily
| constructivist though something described by one of them can be
| explicitly descriptive for some applications. Any approach
| which produces all real numbers as commonly accepted will fail
| to be explicit in all cases as such explicitness presumably
| implies the real number has been expressed uniquely with finite
| strings and finite alphabets which can describe at most a
| countable number of them.
|
| The decimal numbers, for example, can be viewed as an infinite
| converging sum of powers of ten. Theoretically one could
| produce a description, but only a countable number of those
| could be written down in finite terms (some kind of finite
| recipe). So those finite ones could fall in a constructivist
| camp, but the ones requiring an infinite string to describe
| would, as far as I understand constructivism, not fall under
| being constructivist. To be clear, the finite string doesn't
| have other be explicit about how to produce the numbers, just
| that it is naming the thing and it can be derived from that. So
| square root of 2 names a real number and there is a process to
| compute out the decimals so that exists in a constructivist
| sense. But "most" real numbers could not be named.
| ryandv wrote:
| > several of the definitions in the link feel constructivist,
| i.e. they describe constructions of of real numbers.
|
| If you are a constructivist, then you will supply direct proofs
| for your results as you reject indirect proof, proof by
| contradiction, law of excluded middle, and things of this
| nature.
|
| The converse does not necessarily hold. Providing a direct
| construction of an object satisfying the field and completeness
| axioms (e.g. the Dedekind construction) does not necessarily
| mean that one is a constructivist. Indeed, one can use the
| Dedekind construction and still go on to prove many more
| results on top of it that still do rely on indirect proof and
| reductio ad absurdum.
| jostylr wrote:
| 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
| prmph wrote:
| Its interesting. When I first encountered complex numbers when
| starting high school it was very difficult to wrap my head around
| how they could be actual numbers.
|
| I no longer have that problem, ever since I truly understood how
| all numbers are simply abstract tools for reasoning. In a way,
| it's interesting that complex numbers seem more "real" than the
| real numbers themselves.
|
| I remember listening to a radio show where a physicist discussed
| the link between quantum mechanics and complex numbers, and thus
| how they were fundamental to reality [1], whereas we don't know
| whether real numbers actually describe physical reality.
|
| [1] If I remember correctly, one argument was that although a
| common use of complex numbers is an alternative number system for
| making trigonometric/polar calculations simpler, they underpin
| quantum mechanics in a way that cannot be alternatively
| formulated in terms of real number numbers
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