Set-theoretic difficulties concerning normality In her "Lectures on Set Theoretic Topology" Mary Ellen Rudin states at the end of page 5 that "In the case on nomality this is made doubly difficult by the fact that normality is such a second order property that it can often not be decided whether a given topological space is normal or not within the usual axioms for set theory."
What does it means "nomality is such a second order property"? And why this is a particularity of normality? The regularity isn't a second order property?
Is it possible to define a topological space in ZFC such that it is undecidable wheter this space is normal or not?
 A: 
What does it means "normality is such a second order property"?

It means that the statement "$X$ is a normal space" requires non-trivial quantification over the subsets of some set. In particular, for a topological space $(X,\tau)$ with base $\mathcal{B} \subset \tau$, the assertion that  $X$ is normal, is equivalent to the following, second-order, statement,
$$\forall A,B \in \mathcal{P}(\mathcal{B})\, \exists U, V \in \mathcal{P}(\mathcal{B}): \text{ either } (\cup A) \cup (\cup B) \neq X, \text{ or } \quad\quad\quad\quad\quad$$
$$\quad\quad\quad\quad\quad\quad (X\backslash \cup A) \backslash \cup U= \emptyset,\,\,  (X\backslash \cup B)\backslash \cup V = \emptyset, \text{ and } (\cup U) \cap (\cup V) = \emptyset.$$

Is it possible to define a topological space in ZFC such that it is undecidable whether this space is normal or not?

Yes. 
A nice way to do this is to consider the spaces associated with ladder systems on $\omega_1$. 
Definition: 


*

*A Ladder System on a stationary set $S \subset \lim(\omega_1)$ is an $S$-indexed sequence $M=\langle M_\gamma: \gamma \in S\rangle$ of countable subsets of $\omega_1$, such that, $\operatorname{ot}(M_\gamma) = \omega\,$  and $\,\sup (M_\gamma) = \gamma $.

*For each ladder system $M=\langle M_\gamma: \gamma \in S\rangle$, we let $X_M$ denote the topological space defined on the set $A=\omega_1 \times \{0\} \cup S \times \{1\}$, by letting each $(\gamma, 0) \in A$ be isolated and taking as basic open neighborhoods of each $(\gamma,1) \in A$, sets of the form $\{ (\gamma,1) \} \cup B$ where $B \subset M_\gamma \times \{0\}$ co-finite.
As it turns out, whether any of the spaces $X_M$ are normal, is independent of $\mathsf{ZFC}$; for a nice discussion of this result, see the paper,

Balogh, Zoltán; Eisworth, Todd; Gruenhage, Gary; Pavlov, Oleg; Szeptycki, Paul, Uniformization and anti-uniformization properties of ladder systems, Fundam. Math. 181, No. 3, 189-213 (2004). ZBL1051.03034., 

