[FOM] Real Numbers

[Victor Makarov]

>Debates over Benacerraf's problem always seem to lead to someone's 
>something like the following, which Wiman says here:
>
>>The real numbers are not equivalence classes of Cauchy sequences.  They 
>>can be, but they can also be Dedekind cuts, binary numbers, real closed 
>>fields, one dimensional continua, or a number of other things.
>
>I always feel puzzled when I get to that point, however. ...
>Richard Heck
>

It seems that some authors have mixed together entirely different concepts:

1) the concept of "abstract reals" that can be represented by the theory TR 
of the field R of real numbers;

2) the concept of a model of the theory TR;
(each model of TR introduces "concrete reals");

Some well-known (different) models of the theory TR are "Dedekind cuts" and 
"Cauchy sequences".

Victor Makarov


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