[FOM] Remark on Paradoxes

[Harvey Friedman]

On 7/14/06 10:41 AM, "Michael Kremer" <kremer at uchicago.edu> wrote:

> Harvey Friedman provides a three point analysis of the origin of Paradox,
> and claims that Russell's paradox is a case in point.  According to the
> proposed analysis, there must be "some fundamental principles surrounding
> certain concepts that are seen to be essential, and people use them to good
> effect" and "there are some obvious extensions of these principles" which
> turn out to be inconsistent.
> 
> I would appreciate some enlightenment as to what is intended with regard to
> Russell's paradox.  What are the "obvious principles," what are the
> "certain concepts" they "surround" and are "essential" to, and what are the
> "obvious extensions" of those principles which lead to inconsistency?  I
> have my guesses, but I am not sure.  in particular I am not sure where the
> "obvious principles" I am guessing at are "used to good effect."
> 

An analysis of mathematics reveals the following( from the set theoretic
point of veiw):

1. Mathematics makes continual use of comprehension, but the comprehension
is almost always restricted to the form that we now call Separation.
Specifically, one continually forms {x in A: phi(x)}.

2. Obviously the power of Separation in mathematics requires a supply of
sets A that are obtained by means other than separation. I.e., we use at
least some instances of {x: phi(x)}.

3. These ancillary {x: phi(x)} that are used are extremely special and are
in fact close to separation in the sense that the x's are severely
restricted to a given set(s).

4. The most powerful of these are infinity and power set. In the case of
power set, we have {x: x included in A}. So x is severely restricted not
directly as in separation, but indirectly in that the elements of x are
restricted to lie in A.

5. Infinity is not normally viewed as a principle of the form {x: phi(x)},
but rather as something of a different fundamental character. Most
mathematicians think of it in connection with *finite generation". Finitely
generated structures play an enormously essential role in normal
mathematical activity.

6. So, if we make such essential and productive use of {x in A: phi(x)} and
at least certain selected {x: phi(x)}, why not, as Frege did, propose full
{x: phi(x)}? Of course, that is the disaster called Russell's Paradox.

7. From a mathematically, philosophically, and scientifically illiterate
point of view, the general notion of set - where anything whatsoever can be
an element of a set - is not natural. Or at least not nearly as natural as
the concept of time and space, or "better than", where the latter is
transitive and irreflexive. For these, and other, more intuitively natural
concepts, the Russell Paradox does not seem to get off the ground.

Harvey Friedman