I will now try to start setting up a legitimate foundational program surrounding “maximality in set theory”. I am not aware that we have one. If you know of any, please let us know.
I expect philosophers (and other interested parties) to weigh in raising all sorts of issues as I struggle to develop one, and then I should be able to play Ping Pong with further developments taking into account what they say.
I hope there is interest here in seeing such a real time development – and the quality of what comes out in terms of foundational programs of general intellectual interest.
A working environment for stating “maximality principles” or “principles that are inherent in maximality”, or whatever, is the following class of statements:
*) Let be a binary relation on a set . There exists such that some condition holds relating .
We can take the conditions to be purely universal. Thus we have defined a countably infinite class of sentences of set theory.
EXAMPLE. Let be a binary relation on . There exists such that .
We know that this example is provably equivalent to AxC.
Note that if we put a bound on the number of quantifiers over that are allowed, and we don’t allow to be iterated, then we have only finitely many instances, up to tautological equivalence.
Now it is “obvious” that the above Example is the simplest example leading to an equivalent of AxC. But
PROBLEM. State and prove rigorously that the above example is simplest.
Now for a crucial question. Is there an interesting criteria for determining whether an arbitrary sentence of set theory in this family *) of sentences of set theory, represents a legitimate maximality property?
Arguably, ANY sentence in *) that is “true” or “obvious” or “reasonable” represents an instance of maximality. It is saying that given any , there is a certain kind of associated function from into .
I can hear the complaints already, but I am groping around here, as I feel a lot of traction.
CONJECTURE. All instances of *) without nesting of are either a) provable in a weak fragment of ZF; b) refutable in a weak fragment of ZF; c) provably equivalent to AxC over a weak fragment of ZF. Furthermore, this decision is of low computational complexity.
A consequence of this Conjecture is that there are no CONFLICTS. I.e., we have the kind of ROBUSTNESS that we want for a legitimate foundational program.
MORE CONJECTURES. Allow successively broader and broader forms of *), starting with functions and not just relations.
Where is the threshold, where we can code too much and get pathology?
Normally I would just sequester this development until I had the time to fully see what is going on technically, but I am 66 and have much too many other things on my plate. Go have fun!
Philosophers – please complain so that I can make this more interesting. Hopefully we are just getting started.