FOM: re: CH

[Soren Moller Riis]

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re: CH
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Henry Cohn makes a very interesting observation:

> What if we phrase things without using CH at all?  Let X be an
> uncountable well-ordered set with all initial segments countable.
> Define a probability measure by P(A) = 0 if A is countable and
> P(A)=1 if X-A is countable.  Two players independently choose
> elements x and y of X, and the player choosing y wins iff y>x.
> 
> If x is chosen first, then choosing y according to P wins with
> probability 1, since {y : y<=x} is countable and thus has
> probability 0.  (This does not depend on how x was chosen.)
> Similarly, if y is chosen first, then choosing x according
> to P wins with probability 1.
> 
> This seems to me to be exactly the same situation that arose
> in the discussion of CH, so I don't find the earlier argument
> convincing evidence for not-CH.

First a minor technical point: Strictly speaking you do not define
a probability measure because X can be divided into two disjoint 
uncountable sets A,B. Each of these have measure 1, thus the whole
space have measure 2, but was defined to have measure 1.

Anyway what you clearly have in mind is to define the probability
measure on the sigma-algebra generated by all initial segments
(or equivalent from all intervals).

Your example shows that I smuggled in some additional
assumptions: I forgot to specify which probability spaces are 
allowed as part of a winning strategy. 

A probability space U (with a probability measure defined on some 
sigma-algebra) do of course not in general provide us with a  
procedure by which we can pick an element r \in U randomly such 
that the informal probability of a measurable event corresponds to 
the formal probability.

That there is a problem here is perhaps most clearly seen in 
extreme cases: Consider for example the probability measure on R, 
defined to be 1, on all positive numbers, and zero on non-positive 
numbers (the sigma-algebra consists of {empty-set, R_{-},R_{+},R}). 
Only 4 different sets are measurable. It is certainly clear that 
this probability measure does not provide (in any canonical
fashion) a method of actually choosing r \in R randomly (according 
to this measure). 

How can we capture the idea of choosing a real randomly according 
to some continuous probability distribution. 

In my earlier postings I should definitely have stipulated that 
only certain probability spaces actually can function as part of a 
winning strategy. We must require that there exists an (ideal) 
process by which it is possible to pick an element in the 
probability space such that the probabilities corresponds to real 
expectations. 

I am quite a novice in all these questions - And probably
(whatever that means ;-)) most mathematicians I spoke to were 
so as well. So why does there seems to be such universal agreement
that player II cannot win the game
 " ------------------------------------------------- 
   Player I pick real r \in [0,1]

   Player II choose countable set B by a procedure which
   basically involves selecting a real number s randomly.

   Player II wins if r \in B 
   -------------------------------------------------- "
with probability 1 (Player I choose first)? 

Here is a rather naive way of explaining this intuition.

Imagine we are given a fixed countable set S of reals.
Imagine we have a coins C_1,C_2,...one for each natural 
number. Imagine we flip all coins simultaneously.
After having thrown the coins we have produced (in the 
obvious way) the binary decimal expansion of a real 
number s \in [0,1].  

Does it makes sense to say that it is extremely unlikely
s \in S? The measure of S is zero, but does this allows
us to conclude that s \in S with probability zero
(when probability is used as something representing
actual expectations in an ideal stochastic process).  


It seems to me that all what is required to accept the
variant of Chris Freilings argument is that:

Given a countable set S. If s \in [0,1] is being chosen
by the coin flipping procedure then the (informal) probability
(not the formal probability in the sense of measure theory)
that s \in S is 0.

Joe shipman writes:

> Riis's "proof" of CH calls for such treatment!  Soren, can you please
> try to formulate an statement in the language of set theory from which
> your theorem that the result of the game shouldn't depend on the order
> the players move follows?

It seems to me (but I am no expert) that we first need to 
develop a theory (as part of fom?) which can tell us when formal 
probability spaces can be used to generate random elements. Did anyone
try to develop such a theory? 

Soren Riis