First order logic: "consistency," "compactness"?
Consistency: A set $\Sigma\subseteq\text{WFF}$ is consistent iff there is no $\varphi\in\text{WFF}$ such that $\Sigma\vdash\varphi$ and $\Sigma\vdash\lnot\varphi$.
Compactness: A set $\Sigma$ is consistent iff every finite $\Sigma_0\subseteq\Sigma$ is consistent.
I have read through these definitions, however I still feel like I do not understand them. Can someone respond and break them down for me in simpler terms? I feel more confident in my abilities if I understand them to the fullest and their application.
Thank you