The notion of formal proofs 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 longer 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 $ZFC\vdash_{<\omega}\neg Con_{\alpha}(\text{ZFC})$? 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 $ZFC\vdash_{\alpha}\sigma$ or $ZFC\vdash_{\alpha}\neg \sigma$? 

**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 $ZFC\vdash_{<\omega}2^{\aleph_0}\geq\aleph_{1}$ and so $d(ZFC,2^{\aleph_0}\geq\aleph_{1})<\omega$.

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

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

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  [1]: https://i.sstatic.net/Zuuvy.png