[FOM] Countable/counted

William Tait williamtait at mac.com
Sat Aug 22 12:02:59 EDT 2009

Van der Waerden, in the second edition of his _Modern Algebra_, is a  
good example of someone who would have profited by the distinction  
countable-counted. Between the first and second editions (1931 and  
1937, resp.) he got religion and in the preface to the second edition  
states that he tried to avoid as much as possible "any questionable  
set-theoretical reasoning in algebra". He recognized that the result  
did not meet entirely the demands of intuitionism, but claimed (I am  
quoting from the English translation) "I have completely omitted those  
parts of the theory of fields which [sic] rest on the axiom of choice  
and the well-ordering principle ..." In the first chapter, on page 9,  
he asserts:

> The union of a countable set of countable sets is itself countable.
> PROOF: Let the countable sets be denoted by M_1, M_2, ..., and let  
> the elements of M_i be m_{i1}, m_{i2}, ... .


Bill Tait

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