[FOM] RE: FOM New Axioms
montez at fidnet.com
Fri Jun 27 20:10:33 EDT 2003
[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.
Does this principle always give unique results?
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.
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?
Here is one: if X is a set of real numbers, then X is measurable. This
holds for all finite sets X, but when extended to all sets, implies ~AC.
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
I do not think it is very hard. But I expect it to be somewhat unnatural
in most cases. (Whatever that means.)
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?
There are infinitely many I am sure. At each step, we must make a choice
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
I called it a principle - as in guidelines - rather than an axiom or
or some such.
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.
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
(&) If it can be shown that a proof of a popular theorem uses a certain
and mathematicians do not seem willing to give up this theorem, then axiom a
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
by retro-fitting principle (&) to the mathematics of the ancients.
Moreover, if I
understand the plan of Reverse Mathematics properly, the guiding principle
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
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
step for Reverse Mathematics after quite a few details have been gathered
about current results in contemporary mathematics.
More information about the FOM