[FOM] 487:Invariant Maximality/Naturalness
David Roberts
david.roberts at adelaide.edu.au
Wed Mar 21 17:29:50 EDT 2012
Hi,
some feedback, as requested.
On 21 March 2012 12:52, Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> NOTE: There are only finitely many order invariant S contained in Q[0,16]^32.
Do you know how many?
> Another way of looking at a restricted shift function is as given by a mapping H:Q[0,16]^32 into {0,1}^32. The associated restricted shift function T is given by T(x) = x + H(x).
Now this is a very interesting way of putting it. Finitising as much
of the problem as possible is a good move, it helps for combinatorial
applications (about which see later)
> Here Z+up:Q[0,16]^32 into Q^32 is defined by Z+up(x) = the result of adding 1 to all coordinates greater than all coordinates not in Z+.
Why not say Z+up:Q[0,16]^32 into Q[0,17]^32? Or go with the version
Q[0,16]^32 into {0,1}^32
> ***AMBIENT SPACES Q[0,n]^k***
>
> The same results apply except that we need n >= 16 and k >= 32 in order for our methods to show that independence from ZFC kick in.
Are these sharp bounds now? Would they vary for other notions of equivalence?
-----
Now as to 'naturalness'. It was remarked that the ingredients of the
result are all obvious mathematical in nature, rather than logical:
rationals, order equivalence, relations. However I would argue that
the naturalness does not come only from the input to the problem, the
recipe as it were, but what happens afterwards.
Joel Hamkins argues in a recent paper that 'obvious' set theoretic
principles which settle CH undercut themselves, because we know too
much about the various universes where CH holds and where it doesn't.
In essence, something which should be intuitively acceptable turns out
not to be because it doesn't sit with a massive body of knowledge
(whichever way CH falls with respect to it).
I would like to apply the same argument to Friedman's result. It is
made out of mathematical ingredients (mathematical in the sense that
he himself has used it - namely of interest to mathematicians), but
the actual end product seems to be at present an orphan. It is trying
so hard to be mathematical, but because the construction is finely
tailored for the result, it is difficult to see how else it could be
used. The only mathematics that flows from it (at present) is the
result itself. Now I hope the opposite ends up being true, but until
someone makes the forward linkage to non-independence results, this
may wind up being seen as an isolated result--important, but not
connected to mathematical _practice_. People may fall into the
Hamkinsean gambit of saying, "well, that is an independence result
using something almost, but not quite, entirely unlike combinatorics
[say], but it can't be really mathematical, because it leads to large
cardinals and independence phenomena. In my combinatorial world I
don't get independence results."
My hope is that either some sort of simple reduction of the problem
(to something provable) will be shown to be something very clearly
classsically mathematical (other than the starting statements we are
given at present, since Zorn's lemma is again subject to a
Hamkins-style brush-off), or that something very clearly mathematical
will be shown to depend on the construction Friedman has used.
I hope this is helpful insight into how some people may see it:
hopeful, and willing, but needing to be a little more convinced.
All the best,
David Roberts
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