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 $n$. 

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> How do we prove that these two sets of axioms are equivalent (and therefore characterize $w\_1$)?
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Concerning higher Stiefel-Whitney classes, does a set of axioms like 1''-2'' define $w\_k$ for each $k$?