# [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:

f'(x)=f(x)'

Arnon Avron
```