Immediate consequence in Gödels incompleteness paper

Question

In the famous paper, “On Formally Undecidable Propositions of PM”, $c$ is defined as the immediate consequence of $a$ and $b$ if $a$ is the formula $\lnot b \lor c$. How does this relate to the traditional formulation of deduction from implication involving modus ponens?

Answer 1

In the famous paper, On Formally Undecidable Propositions of PM, $c$ is defined as the immediate consequence of $a$ and $b$ if $a$ is the formula $\neg b \vee c$.

How does this relate to the traditional formulation of deduction from implication involving modus ponens?

In other formulation: $\neg b \vee c, \; b\;\vdash\; c$

It's just a restatement of the traditional modus ponens rule, but substituting the equivalence: $$b\to c \quad\equiv\quad \neg b \vee c$$