[FOM] No coding in ZC???

Arnon Avron aa at tau.ac.il
Mon Feb 27 09:48:36 EST 2006

One of the many things I cannot compete with Friedman (and not
the most important of them) is the number of postings he manages
to send every day, and the number of points raised in these postings.
I'll do my best to react, but I shall do so in my limited pace.
In this posting I am going to discuss just one claim repeatedly
made by Friedman. The last time (if I am not mistaken) in his reply
to Weaver on February 21:

> Without a doubt, the right vehicle is ZC = Zermelo set theory
> with the axiom of choice. You do lose some very interesting things, 
> but there is no coding involved. 

No coding involved? Really? So here are some points to think about:

1) There is no question that when core mathematicians speak on
   a triple (a,b,c) they have in mind the set {{a},{a,{{b},{b,c}}}}.

2) Similarly, ask any mathematician what is the sum of
   {{}} and {{},{{}}} and he will tell you in less then a 
   second that it is {{},{{}},{{},{{}}}}
     And speaking about sum, it goes without saying that 
   the symbol + is understood by everybody to refer to a certain
   infinite set of objects of the form {{a},{a,{{b},{b,c}}}}
   where a,b,c are finite von Neumann ordinals.

3) Each positive rational number is (as any pupil in school knows) 
   a certain infinite collection of objects of the form 
   {{{{a},{a,b}}},{{{a},{a,b}},{{c},{c,d}}}} (I hope I have put 
   it right...), where a,b,c,d are von Neumann ordinals. 
     And of course, of course, every core mathematician regularly
   thinks of real numbers (and handles them accordingly) as 
   certain infinite sets of certain infinite sets of objects of
   the form {{{{a},{a,b}}},{{{a},{a,b}},{{c},{c,d}}}} 
   where a,b,c,d are von Neumann ordinals (well, not exactly,
   I forgot to recall what everybody understand by  "rational 
   numbers", including the negative ones. Never mind).

4) Since ZFC is the golden standard of the whole of mathematics,
   it includes of course Geometry too. Yet I suspect that Euclid
   would not have immediately agreed had he been told that no
   coding is involved in seeing points in space as objects of the
   form {{x},{x,{{y},{y,z}}}}, where each of x,y,z is a
   certain infinite set of certain infinite sets of objects of
   the form {{{{a},{a,b}}},{{{a},{a,b}},{{c},{c,d}}}} 
   where a,b,c,d are finite von Neumann ordinals.

5) A crucial case of coding of a central mathematical concept
   which most of core mathematicians *practically* do not accept
   and use is the concept of a function. In set theory a function 
   is just a set of pairs Which satisfies a certain property.
   Officially, core mathematicians might accept this coding.
   Practically they do not. This is obvious from inspecting any
   text on analysis. It is full with propositions of the form:
   "Let f(x) be a continuous function ...". If somebody really
   thinks on functions as sets of pairs then there is simply no 
   meaning whatever to refer to "x" in "the function f(x)"
   (core mathematicians usually remain speechless when I ask them
   whether the function "f(x)" is or is not identical with the
   function "f(y)" ...). Most mathematicians still think of a
   function as a correspondence between a dependent variable
   and independent variables, and speak (in all textbooks) about
   function of one variables, or functions of several variables,
   or of f as a function of the variable x etc.

   Actually, the official coding  of the concept of a function
   in set theory is not respected by the set theorists themselves. 
   Thus in my previous posting I mentioned Jensen's Rudimentary
   functions. These functions are not functions at all according
   to the official coding of functions in set theory, since they 
   are not sets of ordered pairs. Similarly, all the ordinal
   functions are not "functions". So in this case the set-theoretical
   coding does not even fully capture the coded notion.

   We have a similar phenomenon concerning the mathematical notion
   of relation. I suspect that few mathematicians really think
   of the relation < as a set of ordered pairs of reals. What is 
   certain is that no mathematician, not even set-theorists, thinks 
   of the basic relations of set theory: epsilon, =, and \subset
   as sets of ordered pairs (I am having hard time each year in trying
   to explain to students this subtle point).

Some clarifications: First: all the above should NOT be taken as another 
attack on ZFC or ZC. Coding is not necessarily a bad thing. In fact,
I personally find the coding of the natural numbers as finite
von Neumann ordinals as an *excellent* coding. My only point is 
just that it is simply false to claim that ZC is good and 
other systems are bad because ZC does not involve coding while 
the attacked systems do. It is at most a matter of degree.
Third: I agree that it is desirable to reduce the amount of coding 
and to try to make systems as natural as possible. However, I do not
find this to be an essential issue from a philosophical or 
a foundational point of view (but I am interested only in good
old-fashioned FOM, so someone who is guided by new fashions
might feel otherwise).

   One last comment: the case of functions is partially a good example in
which one should not take into account what "core mathematicians" are
doing, but what they should (yes, SHOULD) do. "Core mathematicians"
are horribly confused and careless in their understanding and use 
of variables. Still, they strongly resist the idea that they SHOULD use
Lambda notations for functions. As a result, one might get, e.g., interesting
reactions if one asks some teachers of 
analysis whether the following equation/identity is valid and 
whether their students should understand it as such: 


Arnon Avron

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