Hello! I would like to open a discussion about the Axiom of Symmetry of Freiling, since I didn't find in MO a dedicated question. I'll first try to summarize it, and the ask a couple of questions.

**DESCRIPTION**

The *Axiom of Symmetry*, was proposed in 1986 by Freiling and it states that

$AS$: for all $f:I\rightarrow I_{\omega}$ the following holds: $\exists x \exists y. ( x \not\in f(y) \wedge y\not\in f(x) )$

where $I$ is the real interval $[0,1]$, and $I_{\omega}$ is the set of countable subsets of $I$.

It is known that $AS = \neg CH$. What makes this axiom interesting is that it is explained and justified using an apparently clear probabilistic argument, which I'll try to formulate as follow:

Let us fix $f\in I\rightarrow I_{\omega}$. We throw two darts at the real interval $I=[0,1]$ which will reach some points $x$ and $y$ randomly. Suppose that when the first dart hits $I$, in some point $x$, the second dart is still flying. Now since $x$ is fixed, and $f(x)$ is countable (and therefore null) the probability that the second dart will hit a point $y\in f(x)$ is $0$. Now Freiling says (quote),

Now, by the symmetry of the situation (the real number line does not really know which dart was thrown first or second), we could also say that the first dart will not be in the set $f(y)$ assigned to the second one.

This is deliberately an informal statement which you might find intuitive or not.
However, Freiling concludes basically saying that, since picking two reals $x$ and $y$ at random, we have *almost surely* a pair $(x,y)$ such that, $x \not\in f(y) \wedge y\not\in f(x) )$, then, at the very least, there exists such a pair, and so $AS$ holds.

**DISCUSSION**

If you try to formalize the scenario, you'd probably model the "throwing two darts" as choosing a point $(x,y) \in [0,1]^{2}$. Fixed an arbitrary $f\in I\rightarrow I_{\omega}$, Freiling's argument would be good, if the set

$BAD = ${$(x,y) | x\in f(y) \vee y \in f(x) $}

has probability $0$. $BAD$ is the set of points which do **not** satisfy the constraints of $AS$. If $BAD$ had measure zero, than finding a good pair would be simple, just randomly choose one!
In my opinion the argument would be equally good, if $BAD$ had "measure" strictly less than $1$. In this case we might need a lot of attempts, but almost surely we would find a good pair after a while.

However $BAD$ needs not to be measurable. We might hope that $BAD$ had outermeasure $<1$, this would still be good enough, I believe.

However, if $CH$ holds there exists a function $f_{CH}:I\rightarrow I_{\omega}$ such that $BAD$ is actually the whole set $[0,1]^{2}$!! This $f_{CH}$ is defined using a well-order of $[0,1]$ and defining $f_{CH}(x) = ${$y | y \leq x $}. Under $CH$ the set $f(x)$ is countable for every $x\in[0,1]$. Therefore

$BAD = ${$ (x,y) | x\in f_{CH}(y) \vee y \in f_{CH}(x) $}$ = ${$ (x,y) | x\leq y \vee y \leq x $}$ =[0,1]^{2}$

So it looks like that under this formulation of the problem, if $CH$ then $\neg AS$, which is not surprisingly at all since $ZFC\vdash AS = \neg CH$. Also I don't see any problem related with the "measurability" of $BAD$.

**QUESTIONS**

Clearly it is not possible to formalize and prove $AS$. However the discussion above seems very clear to me, and it just follows that if $CH$ than $BAD$ is the whole set $[0,1]^{2}$. without the need of any non-measurable sets or strange things. And since picking at random a point in $[0,1]^{2}$ is like throwing two darts, I don't really think $AS$ should be true, or at least I don't find the probabilistic explanation very convincing.

On the other hand there is something intuitively true on Freiling's argument.

My questions, (quite vague though, I would just like to know what you think about $AS$), are the following.

A) Clearly Freiling's makes his point, on the basis that the axioms of probability theory are too restrictive, and do not capture all our intuitions. This might be true if the problem was with some weird non-measurable sets, but in the discussion above, non of these weird things are used. Did I miss something?

B) After $AS$ was introduced, somebody tried to tailor some "probability-theory" to capture Freling's intuitions? More in general, is there any follow up, you are aware of?

C) Where do you see that Freiling's argument deviates (even philosophically) from my discussion using $[0,1]^{2}$. I suspect the crucial, conceptual difference, is in seeing the choice of two random reals as, necessarily, a random choice of one after the other, but with the property that this arbitrary non-deterministic choice, has no consequences at all.

Thank you in advance,

Matteo Mio