[FOM] real numbers

[Hartley Slater]

Lucas Wiman (FOm Digest Vol 4 Issue 32) says real numbers are Cauchy 
sequences of rational numbers, and that they are 'equal' if the 
difference between corresponding members gets arbitrarily close to 
zero.  Real numbers, however, are standardly defined as equivalence 
classes of Cauchy sequences of rationals, i.e. as classes of equal 
reals, in Wiman's sense.

Wiman goes on 'Sequential completeness is the fundamental idea of a 
continuum as determined by decades of research in analysis by some of 
the greatest mathematicians of all time in the nineteenth century' - 
some of which he names.  But he, like them, misses the category 
mistake in their definitions, for, whether on Wiman's, or the 
standard definition, real numbers are not comparable to rational 
numbers, and so have no representation on a line.

Taking the standard definition, what may be greater, less than or 
equal to a real number is a 'rational real number' (see, for 
instance, P. Suppes, Axiomatic Set Theory, Blackwell, Oxford 1972, 
pp183, 185), which is the equivalence class of a sequence consisting 
of the same rational number repeated endlessly. But that means that, 
while a rational number (like 1/2, say) can be thought of as a single 
point on a line in physical space, the associated rational real 
number has no such representation, since it cannot be either less 
than, equal to, or greater than the rational number associated with 
it.  It is in a quite different category.  Moreover in Cauchy 
sequence space, where the rational real number does have a place, it 
is not a single point but an interval, with a clear 'inside'.  For 
there are equivalent sequences of rationals, within that real number, 
which are always distinct, i.e. whose corresponding differences are 
never zero.

And that also takes the edge off the supposed 'completeness' of the 
reals.  For while the above equivalence classes partition Cauchy 
sequence space, and a bounded sequence of cells in that partition 
will indeed have a least upper bound which is another such cell, 
*within that 'least upper bound' cell* there will be no Cauchy 
sequence which is least (or greatest) - in the sense that there is no 
other Cauchy sequence in the same cell with corresponding members 
less than it.  That is because of the continual possibility of 
distinctness within the same cell.  You have to think of each cell as 
a single thing, with no 'inside', to think there is a greatest single 
thing which completes the sequence of cells, for there is no greatest 
*Cauchy sequence*, in the above sense, amongst all the Cauchy 
sequences in the cells in the sequence of cells, any more than there 
is a least Cauchy sequence in the cell which 'completes' the sequence 
of cells.  But the idea that a real number is a single thing comes 
from the conflation of rational real numbers with rational numbers. 
It is merely, therefore, that *those* Cauchy sequences in the group 
which has been collectively called 'the' least upper bound are 'least 
upper bounds' of the remaining Cauchy sequences.

It used to be common to write 'All lions are carnivorous' as 'The 
Lion is carnivorous', making out that a single object was involed. 
In particular this was common parlance in the nineteenth century 
which Wiman still honours.  But we have since learnt to discriminate 
amongst lions.
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html