[FOM] Real Numbers

Richard G Heck heck at fas.harvard.edu
Sat May 3 10:10:08 EDT 2003

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. What is the 
modality here? In what sense /can/ the reals be any of these things, 
though they are not? I understand what it means to say, for example, 
that, though I am not a mathematician, I might have been, but that is 
surely not the sort of thing one has in mind here. If I am not a 
mathematician, then perhaps I /could/ be one, but what would it mean to 
say that I /can/ be one? Surely if the reals /are not/ Dedekind cuts, 
then they /cannot/ be Dedekind cuts, any more than I can be George Bush 
if I am not George Bush.

We can interpret analysis in set theory. That is true. But why should it 
mean that the reals "can be" sets?

Now granted, there is a point at which one might be trying to get using 
such language, and it is a point with real mathematical content. But I 
do not see that it has been clearly expressed, either here or elsewhere.

Richard Heck

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