To define mathematical objects, it seems one defines them in terms of other mathematical objects.

However various mathematical objects have different "definitions".

E.g. it seems people "construct" the real numbers (they use objects other than the real numbers, to create an algebraic structure isomorphic to what others consider the real numbers) and then define those as the real numbers. Yet this appears ambiguous to me.

If one were to refer to the "real numbers" would they then be referring to Dedekind Cuts?

Or would they be referring to equivalence classes of Cauchy Sequences?

Or to some other "construction" entirely?

Here are my best two guesses as to what people mean:

$1.$ They are simply referring to any structure isomorphic to the (unique) complete ordered field.

$2.$ They are referring to an equivalence-class$^{*}$ of all structures that are isomorphic to the (unique) complete ordered field.

$*\small(\text{Not an equivalence class in the proper sense with sets as an such object cant be a set by Russell's paradox)}$

The reason $1$ would not result in ambiguity is because we often never need to make use of their original "definitions" per se. Because we can embed the rationals, integers etc. inside the real numbers. However $2$ would avoid this problem entirely as we have only a single definition.

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    $\begingroup$ Dedekind cuts, eq. classes of Cauchy sequences: we get "the same". This in fact is a rather interesting theorem, but no ambiguity there. $\endgroup$ – DonAntonio Apr 4 '18 at 20:47
  • $\begingroup$ Could you add an example illustrating what you claim, e.g., "it seems people "construct" the real numbers (they use objects other than the real numbers, to create an algebraic structure isomorphic to what others consider the real numbers) and then define those as the real numbers." $\endgroup$ – Namaste Apr 4 '18 at 20:53
  • $\begingroup$ "complete archimidean ordered field" $\endgroup$ – Max Apr 4 '18 at 21:47
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    $\begingroup$ @Max... Well, "complete" (in the sense of order, the only sense that is defined for "ordered field") means every nonempty set bounded above has a least upper bound. This implies archimedean. $\endgroup$ – GEdgar Apr 4 '18 at 22:03
  • $\begingroup$ @GEdgar indeed I must have been thinking of something else (I had $\mathbb{Q}_p$ in mind, which is complete but metrically) $\endgroup$ – Max Apr 4 '18 at 22:18

Your question is essentially a philosophical one. When ordinary people talk about number, they have a very definite concept in mind. Mathematicians are ordinary people, but have a professional obligation to give logical justifications for their use of the number concept. So when they are concerned about logical foundations, mathematicians give existence proofs for the various number systems by constructing explicit witnesses for systems satisfying the required properties. When they go back to their day-to-day mathematical work, mathematicians forget about the details of these existence proofs and happily work as if $\Bbb{R}$ were a subset of $\Bbb{C}$ and $\sqrt{2}$ were an object that belongs to $\Bbb{R}$, satisfies $x^2 = 2$, but isn't itself a set or a sequence.

There is a seminal paper by Benacerraf What Numbers Could Not Be that anyone interested in the philosophical foundations of mathematics should read.

  • $\begingroup$ Thanks for the link, also in regards to mathematical platonism discussed in the article. I didn't know this view even had a name, actually I kind of took it as fact. How can it be that mathematical objects are dependent on us? I mean different collections of humans independently created different systems of counting and arithmetic which are isomorphic to each other - but the theorems they derive are not dependent on the particular symbols chosen, and thus any theorems in one system holds if and only it holds in the second system. $\endgroup$ – user3865123 Apr 5 '18 at 4:04
  • $\begingroup$ It feels like arguing "there is no such thing as human language" because some speak English and others French. $\endgroup$ – user3865123 Apr 5 '18 at 4:04
  • $\begingroup$ @user3865123 There is a lot of commonality between humans (biologically) and they are all embedded in the same physical reality. It also took a long time for things like $0$ and fractions to be invented, and there were and are plenty of divergent opinions on more abstract mathematical objects. Even restricting to ZFC, the nominal "foundation" of mathematics, assuming it's consistent at all, there is an infinite number of incompatible extensions of it. Is the Continuum Hypothesis true or false? It's unlikely any physical experiment will let us decide this. $\endgroup$ – Derek Elkins Apr 5 '18 at 5:34

I'd go for (1), slightly modified. If you've just constructed the real numbers in some book or paper, using, say, Dedekind cuts, then in that context a real number is a Dedekind cut, and you use properties of Dedekind cuts to prove theorems about real numbers.

Since there is a unique ordered field (up to unique isomorphism) and your construction provides one, your theorems will be true for anyone else's "real numbers".

Edit: this is essentially @AsafKaraglia 's answer to the duplicate, but much less informative.


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