[FOM] 489:Invariant Maximality Programs

[Dmytro Taranovsky]

Dear Harvey Friedman,

Here are observations related to your program, including naturalness of 
the results.

I think that the program and your results are natural.  For Z+up 
invariance, you would want to explain why this function is important, 
but if you are able to solve the decision problem from large cardinals 
for a large natural class of relations, you can circumvent the question 
about the importance of particular instances.

To attract other mathematicians to the area, you would need one or more 
of the following:
- practical applications
- connections to other areas of mathematics
- beauty of the theory itself

Currently, you have a strong connection to large cardinal axioms, and 
the beauty stems from that connection.  However, one question to ask is 
how much emphasis would your area get but for the connection with large 
cardinal axioms.

Besides naturalness, the key question is whether to treat your 
propositions as theorems (and the necessity of the use of large 
cardinals as reverse mathematics) or as independence results.

Because of the unfortunate general disinterest in foundations of 
mathematics and set theory, most mathematicians do not understand ZFC, 
let alone have opinions on large cardinal axioms.  Without consensus, 
the status quo prevails.  By linguistic convention, provability when the 
axioms are not specified means provability in ZFC.  However, I do not 
think this convention will last forever.  I think that eventually, we 
will have a "theory of everything" for set theory, that is a single 
statement (or in the language of set theory, a schema) that correctly 
resolves all major incompleteness in ZFC.  The statement -- after enough 
theory is developed -- will be natural and intuitively true.  We already 
have that statement for the language of second order arithmetic 
(specifically, projective determinacy) and a bit beyond.  One option for 
the mathematical community is not wait for the "theory of everything", 
and accept projective determinacy as an axiom now, which would make your 
propositions theorems.  However, one argument for waiting is that even 
if ZFC remains the convention, we can still use other axioms as long as 
we mention their use.

While your emphasis is on Pi^0_1 statements, perhaps your results can 
also lead to natural Pi^1_1 statements, and in turn to a natural ordinal 
notation system for ZFC + {n-ineffable cardinal: n a natural number}.  A 
simple ordinal notation system that captures all ordinals canonically 
provably definable in ZFC (plus some large cardinal axioms) may lead to 
a qualitatively new understanding of the theory.  It would also make the 
theory appear concrete, and thus address one major objection against 
infinitary set theory.

As for practical applications of large cardinal axioms, I think they 
will eventually appear, but their current absence does not make the 
study of large cardinal axioms unimportant.  An ideal scenario for 
forcing confrontation about large cardinal axioms would be a new nuclear 
reactor that generates more energy but whose safety was demonstrated in 
part through the use of large cardinal axioms.

Sincerely,
Dmytro Taranovsky