Is it proper to say (as I keep reading) that the real numbers are "equal" to the equivalence classes of Cauchy sequences in the completion of the rational numbers. Yes, there is a one-to-one correspondence with the limits of the sequences, but "equal to" seems too strong. Am I correct in this?

  • $\begingroup$ Yes, this is how real numbers have been constructed. $\endgroup$ – Extremal Dec 5 '14 at 23:47
  • $\begingroup$ So, given a real number $r$ there is a cauchy sequence of rational numbers that converges to $r$. $\endgroup$ – Extremal Dec 5 '14 at 23:50
  • $\begingroup$ But why not say reals are equal to the limit of those sequences? To say a single real number is "equal to" a class of sets seems like an unnecessary mix of apples and oranges. $\endgroup$ – MPitts Dec 5 '14 at 23:52
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    $\begingroup$ Whether they are strictly equal depends on your particular choice of definitions. They are however isomorphic in pretty much every nice sense you can think of (as ordered fields, as metric spaces, ...), regardless of your choice of definitions. $\endgroup$ – Ian Dec 5 '14 at 23:52
  • $\begingroup$ Note that people constructed different types of numbers in the following order. 1)natural numbers 2)integers 3)rational numbers and finally real numbers. After constructing rationals they found that there are sequences of rationals that approaches values which are not rational (today we called them irrationals). This motivates to construct real numbers like this. $\endgroup$ – Extremal Dec 5 '14 at 23:56

There are several possible construction of the real numbers that, in the usual framework of set theory (ZFC or equivalent) give rise to isomorphic structures.

It doesn't matter how the real numbers are constructed, so long as they are an archimedean ordered field $F$ which is Cantor-complete (if $S$ and $T$ are non empty subsets of $F$ such that $s\le t$, for all $s\in S$ and $t\in T$, then there exists $r\in F$ such that $s\le r\le t$, for all $s\in S$ and $t\in T$).

Two such structures are isomorphic in a strong sense: there is a unique field isomorphism between them. In particular they can't be told apart by just using properties of ordered fields. Moreover there is a unique embedding of the rational numbers in such a field.

The fact that one is the set of equivalence classes of Cauchy sequences in the rationals and the other is, maybe, the set of Dedekind cuts of the rationals is completely irrelevant, as far as the theory of real numbers is concerned. One just uses their properties summarized above.

If you like to think that a real number is an equivalence class of Cauchy sequences, you're welcome, but it really adds nothing to your understanding of the real numbers. The constructions are important because with them we know we're talking about something.

Such a field can be given a uniformity which makes it into a topological field (the operations are continuous), so a notion of Cauchy net can be given (that doesn't appeal to the real numbers) and it can be proved that this uniform space is complete, that is, every Cauchy net converges. In particular (using the unique embedding of the rationals mentioned above), every Cauchy sequence of rationals converges.

  • $\begingroup$ Cantor complete? Is that a thing? $\endgroup$ – Asaf Karagila Dec 6 '14 at 7:21
  • $\begingroup$ @AsafKaragila Why not? $\endgroup$ – egreg Dec 6 '14 at 8:37
  • $\begingroup$ No reason, I just don't recall hearing that term before. Dedekind complete, sure. But Cantor complete? $\endgroup$ – Asaf Karagila Dec 6 '14 at 8:38
  • $\begingroup$ @AsafKaragila Dedekind's and Cantor's conditions are slightly different, but equivalent for Archimedean ordered fields. $\endgroup$ – egreg Dec 6 '14 at 8:39
  • $\begingroup$ If I recall my history right, Dedekind gave his constructions via cuts, and Cantor gave his via Cauchy sequences. $\endgroup$ – Asaf Karagila Dec 6 '14 at 8:47

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