Let $E \to X$ be a vector bundle. We can associate to $E$ several invariants: among them are the *Stiefel-Whitney* classes $w_i(E) \in H^i(X;\mathbb{Z}_2)$. These classes may be defined using the axioms:

0. $w_0(E)=1$ and $w_i(E) \in H^i(X;\mathbb{Z}_2)$.

1. $w(f^*E)=f^*w(E)$ for continuous maps $f$ (here $w=1+w_1+w_2+\cdots$ is the total class)

2. $w(E \oplus F)=w(E) \cup w(F)$

3. $w_1(\gamma_1) \neq 0$ for the tautological bundle $\gamma_1$ over $\mathbb{R}P^{\infty}=BO(1)$

In particular, we can rephrase these axioms to recognize the *first* Stiefel Whitney class:

1'. $w_1(f^*E)=f^*w_1(E)$ for continuous maps $f$

2'. $w_1(E \oplus F)=w_1(E) + w_1(F)$

3'. $w_1(\gamma_1) \neq 0$ for a tautological bundle $\gamma_1$ over $\mathbb{R}\mathbb{P}^{\infty}=BO(1)$.

However, in Lawson-Michelsohn's book *"Spin Geometry"* it is stated that in order for a cohomology class $v_1$ to equal $w_1$ we only need to check:

1''. $v_1(f^*E)=f^*v_1(E)$ for continuous maps $f$

2''. $v_1(\gamma_n) \neq 0$ for the tautological bundle $\gamma_n$ over $BO(n)$ for every natural number $n$.

How do we prove that these two sets of axioms are equivalent (and therefore characterize $w_1$)?

Concerning higher Stiefel-Whitney classes, does a set of axioms like 1''-2'' define $w_k$ for each $k$?