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I consider here the Cauchy equivalent classes of Cauchy sequences which I assume relates to the the limits of the sequences. I assume further that the Cauchy sequences involved cannot be described by finite formulated formulas since they would be countable and cannot be in a one-to-one correlation to the real numbers. This is also supported by Wikipedia’s description of Cauchy sequences which gives the alternate description of a real number x as a limit of a series made up by “the successive truncations of the decimal expansion of x”.

The successive truncation definition is obviously also depending of uncountable definitions, so that there is a one-to-one correspondence between the signifier: the definition, and the signified: the real numbers. In essence, it looks very much like the definition of real numbers is done using real numbers. Isn’t that a problem?

Mikael Jensen
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    Not clear... If you consider the Construction of reals from Cauchy sequences, then the Cauchy sequences used are of rationals. – Mauro ALLEGRANZA Dec 04 '19 at 09:08
  • We had a similar question yesterday. Have a look at that, and the marked duplicate links. Maybe that will help a bit. – Arthur Dec 04 '19 at 09:08
  • Regarding countable-uncountable, we have a basic result of set theory that the power-set of a countable infinite set is uncountable. This result can be applied to the "mother of" all countable infinite sets : $\mathbb N$ to conclude that the power-set of $\mathbb N$ is uncountable. – Mauro ALLEGRANZA Dec 04 '19 at 09:13
  • $\mathbb Q$ is also countable infinite and thus the set of all sequences in $\mathbb Q$ has the same cardinality of the power-set of $\mathbb Q$, i.e. is uncountable. Up to now, no reals involved. – Mauro ALLEGRANZA Dec 04 '19 at 09:17
  • @Allegranza The sequense used are from the rational numbers but it seems to me that they are used in a uncountable way. I wonder if the definition is circular. – Mikael Jensen Dec 04 '19 at 09:28
  • @MikaelJensen The fact that there are as many sequences of rationals as there are real numbers doesn't mean that using sequences of rationals is the same as using real numbers. – Arnaud Mortier Dec 04 '19 at 09:31
  • The sequence are uncountable : correct. But this is a fact proven regardless of the existence of reals or not. The fact that reals are uncountable is a consequence of their definition and is in no way part of the definition itself. – Mauro ALLEGRANZA Dec 04 '19 at 10:02
  • While the "countability" of the naturals is "built-in" into the def of countable infinite itself, the result that there are uncountable sets was a (in some sense unexpected) result proven by Cantor. – Mauro ALLEGRANZA Dec 04 '19 at 10:04
  • @Allegranza The truncated series mentioned in the Wikipedia article is one alternative, which looks very much like a real number. The limit is revealed first at infinity – for most (non-constructible) numbers, i.e. they are described by an infinite series of numbers 0 to 9 in a decimal expansion. (They provably all require the axiom of choice although only one alternative is mentioned with this requirement). Any Cauchy series can be decoded easily into a similar series of numbers. It therefore looks to me like numbers described by numbers. – Mikael Jensen Dec 07 '19 at 00:23

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No, the definition of real numbers via Cauchy sequences is not circular as you seem to suggest. It is true that the set of Cauchy sequences of rationals is not countable. However, we don't need it to be countable. We use one set of cardinality $c$ to define another, new, set of cardinality $c$. This is not circular.

Mikhail Katz
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