Reading this there is a proof with

(1) Suppose $2^\kappa = \kappa^+$. Then there exists a bijection $\sigma : \kappa^+ \to \mathcal{P}(\kappa)$. Setting $f : \mathcal{P}(\kappa) \to \mathcal{P}(\mathcal{P}(\kappa))$ defined via $\sigma(\alpha) \mapsto \{\sigma(\beta) : \beta \preceq \alpha\}$, it is easy to see that this demonstrates the failure of Freiling's axiom.

I don't see it easily. I don't see how such a function demonstrates the failure of the axiom - I guess I don't understand the relation in $\sigma$ and the domain and codomain of f are different than [0,1] and countable subsets of the reals.

I also found this and I think I lack the knowledge about countable ordinals and would like some suggestions on resources for that.

(2) $\text{Proof:} (\Rightarrow) $ Suppose the Continuum Hypothesis is true. Then $\text{card}((0,1)) = \aleph_1$. Well-order the elements of $(0,1)$ as $r_1, r_2, r_3, \ldots$ and let $f(r_i) = \{r_j : j \leq i\}$. Then $f$ is in $\mathcal{A}$, and for $j \leq i$, $r_j$ is in $f(r_i)$, while for $j > i$, $r_i$ is in $f(r_j)$. Furthermore, $f(r_i)$ is countable for all $i$ (having cardinality less than $\aleph_1$) $\Rightarrow$ Freiling’s Axiom of Symmetry is false.

I guess it's almost the same thing as (1) although I'd like to know what's the difference and how they translate the domain and codomain of f in the first proof into the functions in Freiling's axiom.

Any insights, explanations, or references would be greatly appreciated.