[FOM] RE: FOM New Axioms

[Matt Insall]

 [Matt Insall (previously)]
 (**)  When possible, sets behave as do finite sets.  Otherwise,
       proofs or new axioms are required to explain why some sets
       stray from behaving as if they are finite.

[Ralph Hartley]
Does this principle always give unique results?

[Matt]
I would say not.  For example, I get from the principle (**) that
the negation of GCH should hold, but someone else might get, as
I did with (***) [Recall: (***)  Infinite sets behave as much like
finite sets as possible, in an asymptotic sense.] that a stronger
axiom should hold which implies ~CH.  In fact, there is a sense
in which (**) is being used in both cases, but the new axiom produced
is not the same in both cases.

[Ralph Hartley]
I can't off hand think of a statement that is true/provable for all finite
sets which, if extended to all sets, implies CH or ~Choice, but can you
prove there isn't one?

[Matt Insall]
Here is one:  if X is a set of real numbers, then X is measurable.  This
statement
holds for all finite sets X, but when extended to all sets, implies ~AC.

[Ralph Hartley]
How hard is it to find a pair of statements that are both true for all
finite sets, but which can't both be true for all sets (including infinite
ones)?

[Matt Insall]
I do not think it is very hard.  But I expect it to be somewhat unnatural
in most cases.  (Whatever that means.)


[Ralph Hartley]
How many statements are there that are provable/true for all finite sets,
but who's extension to all infinite sets is independent of the axioms?

[Matt Insall]
There are infinitely many I am sure.  At each step, we must make a choice
about
how to apply the principle, but like most principles that are not axioms, my
(***)
cannot be applied always without producing a contradiction.  It is for this
reason
I called it a principle - as in guidelines - rather than an axiom or
meta-axiom,
or some such.

[Ralph Hartley]
The more answers to the last question there are, and the harder it is to
find answers to the second to last, the more productive (**) would be.

[Matt Insall]
You have a good point here.  For this reason, we need to have some other
principles to help us choose our axioms.  I am not sure what they should be,
but it seems that one of the principles has been something like the
following:

(&)  If it can be shown that a proof of a popular theorem uses a certain
axiom, a,
and mathematicians do not seem willing to give up this theorem, then axiom a
is
to be adopted.

It seems that this is how the axiom of choice was adopted, and the
properties that are
considered self-evident for finite sets would, I think, fit into a category
distinguished
by retro-fitting principle (&) to the mathematics of the ancients.
Moreover, if I
understand the plan of Reverse Mathematics properly, the guiding principle
there is
something akin to (&):

Find out which set-theoretic axioms are _required_ in the
proofs of theorems in classical and modern mathematics.

However, I am not sure that in Reverse Mathematics, the goal is to provide a
guiding
principle for choosing new axioms.  It merely describes the situation at
hand for several
previously proved theorems.  Providing a guiding principle like (&) may be a
next logical
step for Reverse Mathematics after quite a few details have been gathered
about current results in contemporary mathematics.