[FOM] 489:Invariant Maximality Programs
Dmytro Taranovsky
dmytro at mit.edu
Sat Mar 24 19:42:16 EDT 2012
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
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