[FOM] Criteria for New Axioms

[Dmytro Taranovsky]

What should be the criterion for adding new axioms to set theory and 
mathematics? Here, I suggest -- and elaborate on -- the conjunction of 
the following:

1. The axiom is true beyond a reasonable doubt.
2. The axiom is known to be independent of other axioms.
3. The axiom fills a particular need.
4. The extent and strength of the axiom is canonical and not arbitrary.

While all four criteria must be met, strength in one criterion can be 
used to compensate for weakness in another; different levels of need for 
the axiom permit different levels of uncertainty.

1. To accept an axiom, we need to be confident that it is true. However, 
an axiom need not be obvious, that is it need not be easily seen to be 
true.  What matters is the degree of certainty (say 99.9% confidence 
that it is true); for a given level of certainty, it should not matter 
much whether that certainty comes from a simple or an advanced study.  
Even the axioms of ZFC are far from obvious if given today to an average 
American without a lengthy explanation. Future axioms will likely 
require a much more advanced study than ZFC does.  However, one should 
keep in mind that:
- Those who study a proposed axiom have a tendency to be biased in its 
favor, both because those who believe it true (or plausible) are drawn 
to study it, and because one feels better if the hypothesis one has 
studied and developed turns out to be true.
- As a practical matter, accessibility to a large fraction of 
mathematicians is important for acceptance.
- There is pedagogical and symbolic value in having natural easily 
understood axioms.

2. Independence means both consistency and unprovability.  A high degree 
of certainty is required about the consistency, including ordinarily a 
calibration of the consistency strength.  The unprovability of the axiom 
should be provable (assuming consistency of other axioms) and not merely 
conjectured.  This criterion ensures that statements like "P != NP" fail 
as proposed axioms.

3. Adding axioms is a serious matter, hence the requirement to fill a 
substantial need.  However, it suffices for that need to be for future 
mathematics rather than for the 20th century mathematical practice.

4. An axiom and its aesthetics should stand the test of time, hence the 
need for nonarbitrariness.  It is preferable to add an axiom one time, 
rather through dozens of small additions.  There is also the option of 
using the axioms when necessary, but always noting their use, especially 
for axioms that are likely to be soon superseded by stronger ones.

How does projective determinacy (PD) fare?
1. I think the evidence for PD is overwhelming, but not everyone agrees.
2. PD is unprovable and we have a high confidence in its consistency.
3. PD is needed for a reasonably complete first order theory of real 
numbers.  However, this is mitigated by that most ordinary mathematics 
deals with Pi-1-2 theorems, and ZFC suffices for all but exceptional 
Pi-1-2 and even Pi-1-3 theorems.
4. Nonarbitrariness of PD as a stopping point depends on one's perspective.

Ordinary mathematics uses real numbers, but not (yet) arbitrary sets of 
real numbers.  PD is necessary and sufficient to avoid major 
incompleteness in the first order theory of real numbers.  PD can be 
shown to be true by a sufficiently deep study of real numbers without 
resolving the question of existence of uncountable sets.

On the other hand, from the point of view of set theory as a whole, PD 
asserts (or rather is equivalent to) closure of real numbers under 
M_n^#, but the same closure makes sense for all sets and not just real 
numbers. The right axiom is that PD holds in every generic extension of 
V, equivalently that V is closed under M_n^# for finite n.

Large cardinal axioms, such as a proper class of Woodin cardinals, are 
also appealing, but the extent of the cumulative hierarchy is 
uncertain.  Currently, large cardinal axioms appear the only natural way 
up, but because we know so little about the true theory of the relevant 
levels of V, it is difficult to be sure beyond a reasonable doubt that 
no natural compelling combinatorial principle will in the future 
contradict existence of Woodin cardinals in V (as opposed to existence 
of canonical models with Woodin cardinals).  Perhaps, for example, a 
natural strengthening of the axiom of choice will strengthen Kunen's 
inconsistency result.  The strength of particular large cardinals can 
also vary depending on what other principles are accepted.  For example, 
existence of a proper class of Woodin cardinals becomes significantly 
stronger if one also accepts existence of a strong cardinal.  Also, a 
proper class of Woodin cardinals is perhaps an arbitrary stopping point, 
but then every consistent principle seems to have a natural extension.

Going further, among strong large cardinal axioms, Vopenka's principle, 
as an axiom schema, is natural and elegant, but is beyond current core 
and inner model theory, and thus its consistency is not sufficiently 
certain.  Vopenka's principle has a number of equivalent forms, one of 
which is the following:  For every proper class -- specifically, for 
every formula (with parameters) denoting a proper class -- of set-sized 
binary relations, one is embeddable into another.  Another equivalent 
form is that for every formula with parameters phi, there is 
phi-supercompact cardinal.  For both forms, one can require the formula 
to be without parameters.

The criterion for new axioms may depend on what it means to accept an 
axiom.  Mathematics does not currently have a central body to arbitrate 
proposed axioms, and acceptance is in a sense a continuum rather than a 
binary decision.
* The nominal acceptance indicator is the following: Given a theorem 
stated without specifying axioms, which axioms can be used to prove it?
* Another indicator is the following:  In areas where the axiom is 
helpful, does mathematical practice concentrate on provability using the 
axiom as opposed to provability in ZFC?  In this weaker sense, PD has 
been accepted.  Theorems about projective sets are proved using PD; 
provability of theorems about projective sets in ZFC is a nice-to-have 
bonus, like provability of theorems in analysis in RCA_0, and the 
network of independence results is reverse mathematics.  (That said, 
reverse mathematics is important.)
* One can also imagine -- or with sufficient interest create -- a body 
that decides on axioms, but whose decisions are symbolic rather than 
giving a license to silently omit the axioms in published theorems.  
This way the decisions are reversible without throwing existing theorems 
into doubt.  My view is that PD or a natural strengthening of PD should 
be accepted as an axiom in this symbolic sense.  At present, I am not 
sure whether theorems that use PD should be required to note its use.

While I am a platonist, the search for new axioms is also relevant for 
formalists, as it leads mathematicians towards better (more beautiful, 
useful, etc.) formal theories.  For a formalist, some of the search can 
be viewed as supplying precise definitions for vague terms.  If neither 
CH nor its negation were implicit in our conception of sets, then for a 
formalist, whether CH is true is a bit like whether Pluto is a planet:  
In some sense it is, and in some sense it is not.  One then looks 
whether mathematical practice will be better with CH, or with negation 
of CH, or with CH unsettled.  If accepting GCH will, beyond a reasonable 
doubt, improve mathematical practice, then a pragmatic formalist will 
accept it.

Sincerely,
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm