[FOM] Remark on Paradoxes

[Harvey Friedman]

This is a brief, but possibly important, remark on the Paradoxes.

Paradoxes come about because of the following situation.

1. There are some fundamental principles surrounding certain concepts that
are seen to be essential, and people use them to good effect.

2. There are also some obvious extensions of these principles, whose
conceptual basis is either the same, or quite similar. At least uncomforably
close.

3. However, these extensions are shown to be inconsistent.

The most famous case in point is of course Russell's paradox in set theory
and related contexts.

But what appears to be the case is that for the informal concepts being
analyzed in Concept Calculus, Paradoxes do not seem to be readily available.
I will explain a bit more below.

This state of affairs could be viewed as a consequence of the following:

The concept of set (where sets can be members of sets) is a notion that is
not, by some standards, sufficiently primitive. The same can be said about
variant concepts like property or function, where also properties may apply
to properties and functions may apply to functions.

Under the strategy of Concept Calculus, we take ideas that are far more
familiar to everyone in the general population as primitive - such as time,
space, objects, better than, etcetera.

Set theory becomes the essential measuring tool - like the real number
system employed in the usual calculus (as in undergraduate mathematics).

Set theory rids itself of time, space, etcetera, in favor of immutable
objects, frozen in time and space.

Look at what Russell's Paradox could mean for, say, just one of the topics
in Concept Calculus - a Varying Quantity.

One could try to write down a Russell Paradox in this context, but it just
doesn't make ANY conceptual sense. Or at least, it involves a major
conceptual shift. 

Recall that the motivating principle behind the Existence axiom used in
Varying Quantity is Aristotle's version of the Axiom of Plenitude
(distinguished from other Axioms of Plenitude): anything that can happen
will. 

In Russell's Paradox in set theory, the word "Paradox" is used because there
seems to be no major conceptual shift in going from the accepted Separation
Axiom to Full Comprehension Axiom. Yet the former is good and the latter is
worse than bad. 

My point is even clearer in the case of the Concept Calculus 5 work on
Better Than (to appear). Here Better Than is an irreflexive transitive
relation. Existence takes the form

"for any bounded range of things, there is something that is better than all
of them, and nothing else."

Now, look what happens if you even TRY to make a Paradox by extending this:

"for any range of things, there is something that is better than all of
them, and nothing else".

But this is absurd on the face of it, and of course, counterintuitive, since
nothing can be better than everything, by irreflexivity. Even the
mathematically and philosophically and scientifically illiterate (whose
thinking is a motivating target of analysis for Concept Calculus)
immediately see that the above is absurd, on the face of it. Yet it took
Russell to tell Frege (a truly great mind, credited with the predicate
calculus) where he went wrong!!!!!!!!

So this might, in retrospect, be a clear indication that there is perhaps
something unnatural about the abstract notion of set which allows - and
lives off of - the idea that a set can have any sets as elements.

Of course, I am convinced that set theory IS the natural MEASURING TOOL for
concepts. After all, we want our measuring tools to be engraved in stone,
immutable, and divorced from the phenomena being analyzed. We want it to
have something related to - but by no means close to - the phenomenon under
investigation. 

The real number system does exactly this for traditional Mathematical
Science.  

However, the informal notions of everyday life and naive thinking seem to
have fundamental properties that cannot be directly modeled in the real
numbers. The properties are highly nonseperable. In fact, they cannot be
modeled in mathematics - without going to set theory well beyond ZFC in many
contexts.

Harvey Friedman