[FOM] "Mathematician in the street" on AC

[Arnon Avron]

Vaughan,

I am afraid that this time you simply contradict yourself.

On the one end you define a countable set to be "a pair (X,f) 
consisting of a set X and an  injection f: X --> N". 
On the other hand you present e.g. Wachsmut as somebody
who uses your definition (rather than the one found
in any textbook). But what Wachsmut says is (according
to you!): " `The proof of the  next statement - that the 
countable union of countable sets is again  countable - 
is very similar' [to the proof that N^2 is countable]".
Now according to your definition it is *meaningless* to
say that N^2 is countable! All you can say are things
like "(N^2,+) is not countable" or "(N^2,\lambda i,j. 2^i*3^j)"
is countable... 

With the exception (perhaps) of particularly
devoted constructivists, no mathematician speaks
or thinks according to your definition. This includes *you*. 
Thus you continue:
 "Given a countable family (X_i,f_i) of 
 countable sets indexed by natural numbers i, 
 construct an injection  q: U --> N where U 
 is the union of the family as follows"
Well, "in what followed" U was the union of the X_i's, not the 
union of the countable family (X_i,f_i) (so
presumably you change also the meaning of the
word "union", not only "countable"?). More important:
according to your own definition you are 
saying meaningless words when you
speak about the "countable family (X_i,f_i)".
All you can speak about according to your
definition of a "countable set" as a pair
is the "countable set" ({(X_i,f_i): i\in N},f), where f is some
injective function from the set {(X_i,f_i): i\in N} to N!

To sum up: my impression is that when you talked in your posting
about the "countable family (X_i,f_i)" and about
N^2 being countable you were not using your own definition,
but the usual definition. I hope that this observation
will convince you (as you have asked at the
end of your posting) that the usual definition does
have some advantages. 

Arnon