[FOM] New Axioms(?)

[Harvey Friedman]

Reply to Ellsworth 3:11PM 6/26/03.

>At each of the last two set-theoretic topology conferences I have
>attended, there have been informal discussions over snacks and beer about
>Woodin's work and whether or not we (the mathematical community) have
>finally been presented with a combinatorial principle that will in time be
>considered as "obviously true" as the other axioms of ZFC.

Even the axiom of choice is not as obviously true as the other axioms
of ZFC. Even after all of this time and all of this work, it seems
virtually impossible to defend the axiom of choice this strongly.

On the other hand, one can try to play revisionist history, and claim
that the axiom of choice was as obvious as the other axioms of ZFC,
but people did not realize that it was, including Cantor, who felt
the need to prove it from the other axioms.

My own opinion is that the axioms of ZFC have a special kind of

coherent simplicity

that cannot be extended. I.e., ZFC is complete.

And also the axioms of ZF have a related special kind of

coherent simplicity

that also cannot be extended. I.e., ZF is complete.

However, it is one thing to say this, and another to make this idea
clear, backed by theorems.

>I am interested in people's opinions about how a "new axiom" gains
>acceptance. [I guess this is asking for the mathematical version of "how
>does a bill become a law?".]

We don't have that many examples to learn from.

With the axiom of choice, we have the following phenomena:

1. It is trivial to see what it says in the context of set theory.
2. It is easy to use.
3. It is useful for general treatments of important subjects. E.g.,
for the general theory of fields, for the general theory of topology
(general topology), for general algebra of various kinds (general
theory of groups), etc. In these subjects, very little can be proved
without the axiom of choice.
4. It is not useful for concrete treatments of important mathematical
subjects. I.e., continuous functions between Polish spaces, theory of
finite field extensions of the rationals, semialgebraic geometry in
the reals, algebraic geometry in the complex numbers, number theory,
etc. In particular, the uses of the axiom of choice in these contexts
are easily removed. But not in 3.
5. There are very simple fundamental sounding philosophies that suggest it.

With 5, there is the simple fundamental sounding philosophy that
suggests the axiom of choice:

*anything imaginable whatsoever, subject to the well known size
limitation, forms a set.*

In particular, a set which meets every one of a set of pairwise
disjoint nonempty sets, is at least IMAGINABLE, and therefore, by
this principle, exists.

It appears that we have run out of fundamental sounding philosophies
of set theory that seem compelling to working set theorists.

Without making any evaluations, at present Woodin's plans for
"proving" the negation of the continuum hypothesis are not
unanimously supported among leading set theorists. Steel 5/19/03
12:47AM writes:

"Woodin goes on to argue that a proof of the Omega-conjecture would
constitute evidence for -CH, but I disagree here. His argument is that
there are Omega-consistent theories which are Omega-complete for the
theory of L(P(omega_1)) (such as the Pmax axiom saying that L(P(omega_1))
is a Pmax extension of L(R)), while by the theorem above, any such theory
would have to contain -CH, as otherwise all Sigma^2_3 sentences would be
Omega-decided by it (since CH implies the quantifier over sets of reals
amounts to a quantifier over subsets of omega_1.) To my mind, however, the
theorem quoted above just says Omega-completeness is not an appropriate
general criterion for theory choice, and so I don't see the logic in then
using it to choose a theory of L(P(omega_1)). (The Pmax axiom does not
give a new domain of conditionally generically absolute statements here,
since it makes the theory of L(P(omega_1)) part of the theory of L(R).)

So in my view, a proof of the Omega-conjecture means we are pretty far
from deciding CH."

However, there may be axioms that become accepted by mathematicians,
even though the fundamental sounding philosophy behind them has
serious weaknesses. This can happen if the applications of them are
strong, in compensating senses.

However, really strong compensating strengths on the application
front, in my opinion, must come in connection with 4 above: i.e., in
the concrete parts of mathematics. In particular, we need uses of
axioms that mathematicians can feel comfortable with using, in the
concrete parts of mathematics, which cannot be removed in favor of
just using ZFC.

This is already happening with Boolean Relation Theory, and there is
a clear plan to develop this subject so that it becomes a clearly
recognizable and well respected part of mathematics that is deeply
connected with an enormous variety of respected mathematical contexts.

Many of you have heard less about Boolean Relation Theory recently
since its main proponent has been very busy with some other matters
in recent months. But that will soon change.

The plan to use new axioms in an essential way for concrete parts of
mathematics seems to be directly connected only with large cardinal
axioms, and not directly connected with issues surrounding, e.g., the
continuum hypothesis and forcing. In particular, this plan will not
shed any light on the continuum hypothesis.

Harvey Friedman