[FOM] Choice of new axioms 1
Harvey Friedman
friedman at math.ohio-state.edu
Tue Feb 14 02:43:30 EST 2006
My reply to Shipman's
http://www.cs.nyu.edu/pipermail/fom/2006-February/009776.html
is my http://www.cs.nyu.edu/pipermail/fom/2006-February/009777.html
which appears earlier than Shipman's. I call this to your attention.
Also, Shipman posted
http://www.cs.nyu.edu/pipermail/fom/2006-February/009772.html
concerning credit for the unprovability of RVM (real valued measurables)
from ZFC. I agree with the historical points in this posting.
There is a new posting of Shipman
http://www.cs.nyu.edu/pipermail/fom/2006-February/009792.html to which I
reply.
************************
>Again, I am not claiming that the validity of RVM depends in any way on
>obsolete intuitions. All I am claiming is that it USED to be the case
>that RVM was considered intuitively plausible,
This is not how I read this history.
There is probably evidence that by "set of reals" people meant "good set of
reals". Then RVM is provable in weak fragments of ZFC.
In fact, the right way to look at this historically is in terms of the
evolution of the notion of "set of reals" - not the evolution of intuitions
about measures. And I doubt that this historical evolution of the notion of
"set of reals" has much to do with physics.
What is the point of these detailed counterfactuals concerning the history
of measure theory and physics? I doubt if it sheds any light on the merits
and demerits of RVM and variants as new axioms for mathematics. Given what
we now know, the existence of a countably additive probability measure on
set theoretically arbitrary subsets of [0,1] is not even close to being
sufficiently attractive for the mathematical community to adopt. Nor do I
see any other plausible historical path which would lead it to be adopted.
To be sure, there is a path to take "set of reals" to mean something
different, like describable, or something, where one rejects that the reals
are well ordered. But then one is in the context of set theory without
choice, with a totally different view of set theory, and the existence of
this probability measure does not have anything like the logical power of
RVM in the usual set theory.
I remind you that RVM does very little for the standard published problems
in the projective hierarchy not resolved by ZFC (essentially nothing at
level 3 and higher), and almost nothing for the standard published problems
in arbitrary subsets of complete separable metric spaces. This is in stark
contrast with the convention V = L, which completely clears out all of the
standard published problems of both kinds.
In my response to Shipman I said that very weak fragments of second order
arithmetic are much more than sufficient to handle all of the standard
physical theories. This is very well known.
Shipman wrote:
>It is not incorrect. The systems they are formulated in may be weak
>LOGICALLY, but as I said, they are not weak ONTOLOGICALLY. You may
>choose to code the functions, Hilbert space operators, and other
>objects of higher type in second-order arithmetic, but the resulting
>theory would be unnatural and unsuitable for actually doing physics.
>When I said "no reformulation...has been plausibly proposed" I meant a
>reformulation that a PHYSICIST would find plausible.
There are a number of conservative extensions of very weak fragments of Z_2
that avoid this coding, and that are totally satisfactory. All of this is
well known. E.g., people calling themselves predicativists (and others) know
this very well, and they can respond to this point if they wish.
It is more delicate to give such conservative extensions of EFA =
exponential function arithmetic, a very weak fragment of PA of special
importance for f.o.m. But there is no doubt that this can be done fully
satisfactorily.
Harvey Friedman
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