# FOM: Elaboration on not-CH proof:

Soren Moller Riis smriis at daimi.aau.dk
Mon Sep 14 14:01:21 EDT 1998

```Elaboration on not-CH proof:
-----------------------------------

Yesterday I presented a variant of a proof of not-CH
[Riis, September 13, 1998]. The argument was a variant of
a wellknown argument by Chris Freiling. Off the list I
have been asked if not my argument was dubious. The following
question was put forward:

> Suppose that Player I and Player II use the same method to
> pick their reals r and s.
> Then the situation is symmetric.   You argue, in effect, that r<s with
> probability 1.  But the same argument also shows that s<r with
> probability 1.  So I guess the notion of picking a real according
> to a probability distribution does not make sense.   Are you then
> concluding that this not making sense implies not CH?

Let me clarify this point:

I was actually very careful the way I defined the game. It is
no coincidence that I let player I makes the first choice!! Suppose
that player II make the first choice (by choosing a countable set
B). Then it follows (even without using CH) that player I by
selecting r randomly (according to any non-singular probability
distribution) will win with probability 1.

I defined the game such that player I must choose first.
It seems to be plain that this should not change the odds in the
game. However one might suggest that the game is not well-defined
(like the game where two players (independently of each other) have
to try to select the largest integer).

The point I am making is that CH implies that NOT only is the
game well-defined [when we carefully demand that player I choose
first], but player II actually has a strategy which guarantee
victory with probability 1.

In short: If we accepts CH, we have to accept that

(1) The game [as I defined it] is well-defined

(2) The related game [where player II choose first] is also well
defined, but the expected outcome is totally different.

The underlying principle is the following:

Suppose we are given a mathematical proposition A, as well as two
experts (players) which are arguing about the validity of A.

One expert (player I) supports A, while the other expert (player II)
supports not-A. The experts communicates via a referee. If one
expert always can persuade the referee she is right (i.e. are capable
to win the game) with some frequency p (say 80%), then there same
frequency p will appear irrespectively of the order by which the