# FOM: re: CH / Funny subsets of RxR

Soren Moller Riis smriis at daimi.aau.dk
Thu Sep 17 16:31:28 EDT 1998

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re: CH / Funny subsets of RxR
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Stephen Fenner writes,

> Cohn's and Schlottmann's analyses of Riis's "proof" of not-CH are
> well-taken and correct; there is no proof of not-CH here.  However, Riis
> it seems has rediscovered one of those pathalogical sets whose existence
> owes both to CH and AC, namely, a (nonmeasurable) subset Z of [0,1]x[0,1]
> such that:
>
> forall x, { y | (x,y) in Z } is co-countable (hence measure 1)
> forall y, { x | (x,y) in Z } is countable (hence measure 0)
>
> In Riis's game, Player I chooses x and Player II essentially chooses y.
>
> There are lots of other subsets of Euclidean space having weird,
> counterintuitive measure properies if we assume AC and CH, and many of

Sorry, but I did not rediscover one of those pathological sets ...
I knew why these set existed before I even stared at university. I
even have constructed some fractal-like structures which besides being
non-measurable have other pathological properties. As part of my
qualifying examen (before I went to study for my PhD at Oxford) I
did work on Banarch-Tarskis paradox. And I do NOT find non-measurable
sets counterintuitive.

When I say I am a novice in this area I didn't mean to say I am not
familiar with the classical ideas from topology and abstract
measure theory. Just that I am a novice when it comes to thinking
about the more philosophical aspects of these questions.

The traditional answer (that the graph not is measurable) is in my
opinion hardly relevant to anything I have said. Though Schlottmann.
Fenner and Cohn might disagree.

In my analysis non-CH must be accepted provided one accepts that it
is possible to select a real number such that any fixed
countable set B is being hit with a zero-probability which is
so-to-speak empirical testable.

Key question:

Fix a countable set B. Can an ideal expert select a real number r
such that r \in B becomes a so-to-speak real and empirical
testable event which happens with neglectable probability?

This meta-mathematical question seems to be outside ordinary
scope of ZFC.
If ZFC+CH is given, it is clear (see example 3 and 4
[Riis, Wed, 16 Sep 1998 11:57:08]) that the answer is negative
i.e. the expert cannot make such a random choice (where a
so-to-speak real empirical testable outcome is created).

Certainly B has measure 0, but this is in some sense irrelevant.
In my first postings I made the mistake of not emphazing the
crucial distinction between formal probabilities (as they are defined
in measure theory) and probabilities as expectations of events
related to randomly chosen elements.

What seems to be the crucial point (in the non-CH argument) is
that an ideal expert (who we can imagine lives somewhere in the
set theoretic universe) can select a real number r such that
r \in B happens, when repeated, with a frequency
which is neglectable.

Soren Riis