The notion of formal proof is defined by finite sequences ($<\omega$ - sequences) of sentences. In some sense if a sentence $\sigma$ is (finitely) provable from the theory $T$ it is very "near" to the "home theory $T$". In this meaning a (finitely) consistent theory is an "origin" with no "near" inconsistency. Also one can think about possibility of "reaching" to an unprovable sentence by increasing the length of the proofs. Now let's generalize the notion of a finite proof to proofs with the length $\alpha$ which $\alpha$ is a (suitable) infinite ordinal, i.e.

Definition 1: We say a sentence $\sigma$ is $\alpha$ - provable form $T$ ($T\vdash_{\alpha} \sigma$) if there is an $\alpha$ - sequence of sentences which includes $\sigma$ such that each member of this sequence is a tautology or an axiom of $T$ or a sentence which is obtainable by deduction rules from former sentences in the sequence.

Remark 1: Each $\alpha$ - provable sentence from $T$ is $\beta$ - provable form $T$ for all $\beta \geq \alpha$.

Definition 2: A theory $T$ is $\alpha$ - consistent ($Con_{\alpha}(T)$) if there is no sentence $\sigma$ such that both $\sigma$ and $\neg \sigma$ are $\alpha$ - provable from $T$. In the other words there is no "inconsistency danger" in the radius $\leq\alpha$ from the origin $T$. Also using $<\alpha$ - sequences we can define $<\alpha$ - consistency of $T$ in the same way.

Remark 2: Each $\alpha$ - consistent theory $T$ is $\beta$ - consistent for all $\beta \leq \alpha$.

Definition 3: A theory $T$ is "totally consistent" if it is $\alpha$ - consistent for arbitrary large $\alpha$.

Remark 3: By Godel's incompleteness theorem $\text{ZFC}$ doesn't imply that $\text{ZFC}$ is $<\omega$ - consistent i.e. $\text{ZFC}\nvdash_{<\omega} Con_{<\omega}(\text{ZFC})$. Also we didn't find any inconsistency "near" $\text{ZFC}$ yet. But it seems possible to find an inconsistency if we go "too far" from $\text{ZFC}$.

Question 1: Is there an ordinal $\alpha \geq \omega$ such that $\text{ZFC}\vdash_{<\omega}\neg Con_{\alpha}(\text{ZFC})$? In the other words is there any finitary proof for $\alpha$ - inconsistency of $\text{ZFC}$ from $\text{ZFC}$?

Question 2: Are there two ordinals $\alpha , \beta \geq \omega$ such that $ZFC\vdash_{\beta}\neg Con_{\alpha}(\text{ZFC})$? In the other words is $\alpha$ - inconsistency of $\text{ZFC}$, $\beta$ - provable from $\text{ZFC}$?

Question 3: Is any unprovable sentence somewhere decidable? Precisely: let $\sigma$ be an unprovable sentence from $\text{ZFC}$ i.e. $\text{ZFC}\nvdash_{<\omega}\sigma$, is there a large enough ordinal $\alpha$ such that $\text{ZFC}\vdash_{\alpha}\sigma$ or $\text{ZFC}\vdash_{\alpha}\neg \sigma$?

Definition 4: Let $T$ be a theory and $\sigma$ a sentence. Define the "distance of $\sigma$ from $T$" as follows: $d(T,\sigma):=min\{\alpha\in Ord~|~T\vdash_{\alpha}\sigma\}$ and if $\{\alpha\in Ord~|~T\vdash_{\alpha}\sigma\}=\emptyset$ define $d(T,\sigma)=\infty$.

Remark 4: We have $\text{ZFC}\vdash_{<\omega}2^{\aleph_0}\geq\aleph_{1}$ and so $d(\text{ZFC},2^{\aleph_0}\geq\aleph_{1})<\omega$.

Question 4: What are the values of $d(\text{ZFC},\text{CH}),d(\text{ZFC},\neg \text{CH})$?

Remark 5: The following (imaginary) diagram says that $d(\text{ZFC},\text{CT})<\omega<d(\text{ZFC},\neg \text{CH})<\alpha<d(\text{ZFC},\text{CH})<\beta$ ($\text{CT}$ means Cantor's theorem). So $\text{ZFC}$ is not totally consistent and if one goes too far from it he/she falls in an inconsistency trap!