In a recent question I was asked to prove the principle of transfinite induction for ordinals but I mistakenly proved the principle of transfinite induction, since I have a only a vague understanding of them, I was just wondering whether there is a key difference between the two or would a proof of the latter be acceptable to obtain all the marks. Thank you for any help.
Principle of transfinite induction: $\forall{x}<{z}(\forall{y} < x[\Phi(y)\rightarrow \Phi(x)]\rightarrow\forall{x}<z\Phi(x))$
Principle of transinite induction for ordinals; $C \neq \phi, \alpha,\beta\in{C}\forall{\alpha}\exists\beta[\alpha \leq \beta]$
@user200632 Your transfinite induction principle is misstated. It ought to be: $$ \forall x\in dom(\Phi))\, \big([(\forall y < x)\,\Phi(y)\to \Phi(x)] \big) \to (\forall x\in dom(\Phi))\, \Phi(x) $$ for well-ordered $\Phi$ (in fact, for well-founded $\Phi$).
Your TI for ordinals is messed up: first you have $\alpha, \beta\in C$, then you quantify over them. Do you mean $(\forall\alpha \in C)(\exists \beta \in C)\, \alpha\le\beta$ ? That is trivially true (take $\beta = \alpha$) even for $C=\emptyset$, and you should get no points whatsoever for proving it :) It should be: $$ (\exists\alpha \in C)(\forall \beta \in C)\, \alpha\le\beta \tag{$\emptyset \ne C\subseteq\sf{Ord}$} $$