(S)WVP - Vopenka's priciple

Question

I would like to understand why precisely - in the nice result here on Vopenka's principle - on the 2nd page, in the last but one paragraph it is enough that there is an ordinal $\kappa$ and a homomorphism $G(\kappa)\to G(\kappa+1)$? I think that I'm missing something very basic.

Answer 1

SWVP says, that the equality $Hom(G(\alpha),G(\alpha'))=\emptyset\Leftrightarrow \alpha<\alpha'$ doesn't hold for any sequence of graphs. He assumes, that the sequence of graphs fulfills $"\Rightarrow"$ in the definition of SWVP. Then you need to show, that $"\Leftarrow"$ doesn't hold, that means, there are $\alpha<\alpha'$. s.t. there is an homomorphism $G(\alpha)\to G(\alpha')$. He shows that there is an $\kappa$ s.t. there is an homomorphism $G(\kappa)\to G(\kappa+1)$.