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 replace the notion of first order logic ($L_{\omega,\omega}$) which uses finite proofs with some infinitary logic like $L_{\alpha,\beta}$ and look at the the theory $\text{ZFC}$ using this new glasses.

Definition 1: A theory $T$ in logic $L_{\alpha , \beta}$ is "totally consistent" if for all ordinals $\gamma\geq \alpha , \delta\geq \beta$ the theory $T$ is consistent in logic $L_{\gamma,\delta}$ too.

Question 1: Assuming consistency of $\text{ZFC}$ in first order logic, is the $L_{\omega , \omega}$ - theory $\text{ZFC}$ totally consistent? In the other words: is it possible to prove an inconsistency from $\text{ZFC}$ by allowing proofs in higher length in infinatry logics?

Question 2: Is any unprovable sentence from $\text{ZFC}$ somewhere decidable? Precisely: let $\sigma$ be an unprovable sentence from $\text{ZFC}$ in logic $L_{\omega ,\omega}$. Are there two large enough ordinals $\alpha , \beta$ such that in the logic $L_{\alpha,\beta}$ there is a proof from $\text{ZFC}$ for $\sigma$ or $\neg \sigma$?

Definition 2: Let $T$ be a first order theory and $\sigma$ a sentence. Define the "distance of $\sigma$ from $T$" as follows: $d(T,\sigma

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 replace the notion of first order logic ($L_{\omega,\omega}$) which uses finite proofs with some infinitary logic like $L_{\alpha,\beta}$ and look at the the theory $\text{ZFC}$ using this new glasses.

Definition 1: A theory $T$ in logic $L_{\alpha , \beta}$ is "totally consistent" if for all ordinals $\gamma\geq \alpha , \delta\geq \beta$ the theory $T$ is consistent in logic $L_{\gamma,\delta}$ too.

Question 1: Assuming consistency of $\text{ZFC}$ in first order logic, is the $L_{\omega , \omega}$ - theory $\text{ZFC}$ totally consistent? In the other words: is it possible to prove an inconsistency from $\text{ZFC}$ by allowing proofs in higher length in infinatry logics?

Question 2: Is any unprovable sentence from $\text{ZFC}$ somewhere decidable? Precisely: let $\sigma$ be an unprovable sentence from $\text{ZFC}$ in logic $L_{\omega ,\omega}$. Are there two large enough ordinals $\alpha , \beta$ such that in the logic $L_{\alpha,\beta}$ there is a proof from $\text{ZFC}$ for $\sigma$ or $\neg \sigma$?

Definition 2: Let $T$ be a first order theory and $\sigma$ a sentence. Define the "distance of $\sigma$ from $T$" as follows: $d(T,\sigma):=min\{\alpha\in \text{Ord}~|~\text{In the logic}~L_{\alpha,\alpha}~\text{there is a proof from}~T~\text{for}~\sigma\}$ and if $\{\alpha\in \text{Ord}~|~\text{In the logic}~L_{\alpha,\alpha}~\text{there is a proof from}~T~\text{for}~\sigma\}=\emptyset$ then $d(T,\sigma)=\infty$

Remark 1: We have $d(\text{ZFC},2^{\aleph_{0}}\geq\aleph_{1})\leq \omega$.

Question 3: What are the values of $d(\text{ZFC},\text{CH})$ and $d(\text{ZFC},\neg\text{CH})$?

Remark 2: The following imaginary diagram shows that $d(\text{ZFC},\text{CT})\leq \omega \leq d(\text{ZFC},\neg\text{CH})\leq \alpha\leq d(\text{ZFC},\text{CH})\leq \beta$ which $\text{CT}$ is Cantor's Theorem. Also it shows that $\text{ZFC}$ as a first order theory is not totally consistent and one reachs to an inconsistency if he/she goes too far from it.

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 replace the notion of first order logic ($L_{\omega,\omega}$) which uses finite proofs with some infinitary logic like $L_{\alpha,\beta}$ and look at the the theory $\text{ZFC}$ using this new glasses.

Definition 1: A theory $T$ in logic $L_{\alpha , \beta}$ is "totally consistent" if for all ordinals $\gamma\geq \alpha , \delta\geq \beta$ the theory $T$ is consistent in logic $L_{\gamma,\delta}$ too.

Question 1: Is the $L_{\omega , \omega}$ - theory $\text{ZFC}$ totally consistent? In the other words: is it possible to prove an inconsistency from $\text{ZFC}$ by allowing proofs in higher length in infinatry logics?

Question 2: Is any unprovable sentence from $\text{ZFC}$ somewhere decidable? Precisely: let $\sigma$ be an unprovable sentence from $\text{ZFC}$ in logic $L_{\omega ,\omega}$. Are there two large enough ordinals $\alpha , \beta$ such that in the logic $L_{\alpha,\beta}$ there is a proof from $\text{ZFC}$ for $\sigma$ or $\neg \sigma$?

Definition 2: Let $T$ be a first order theory and $\sigma$ a sentence. Define the "distance of $\sigma$ from $T$" as follows: $d(T,\sigma):=min\{\alpha\in Ord~|~\sigma~\text{is a theorem of}~T~\text{in logic}~L_{\alpha,\alpha}\}$ and if $\{\alpha\in Ord~|~\sigma~\text{is a theorem of}~T~\text{in logic}~L_{\alpha,\alpha}\}=\emptyset$ define $d(T,\sigma)=\infty$.

Remark 1: We have $d(\text{ZFC},2^{\aleph_{0}}\geq \aleph_1)\leq\omega$.

Question 3: What are the values of $d(\text{ZFC},\text{CH}),d(\text{ZFC},\neg \text{CH})$?

Remark 2: The following (imaginary) diagram says that $d(\text{ZFC},\text{CT})\leq\omega\leq d(\text{ZFC},\neg \text{CH})\leq\alpha<d(\text{ZFC},\text{CH})\leq\beta$ ($\text{CT}$ 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!

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 replace the notion of first order logic ($L_{\omega,\omega}$) which uses finite proofs with some infinitary logic like $L_{\alpha,\beta}$ and look at the the theory $\text{ZFC}$ using this new glasses.

Definition 1: A theory $T$ in logic $L_{\alpha , \beta}$ is "totally consistent" if for all ordinals $\gamma\geq \alpha , \delta\geq \beta$ the theory $T$ is consistent in logic $L_{\gamma,\delta}$ too.

Question 1: Assuming consistency of $\text{ZFC}$ in first order logic, is the $L_{\omega , \omega}$ - theory $\text{ZFC}$ totally consistent? In the other words: is it possible to prove an inconsistency from $\text{ZFC}$ by allowing proofs in higher length in infinatry logics?

Question 2: Is any unprovable sentence from $\text{ZFC}$ somewhere decidable? Precisely: let $\sigma$ be an unprovable sentence from $\text{ZFC}$ in logic $L_{\omega ,\omega}$. Are there two large enough ordinals $\alpha , \beta$ such that in the logic $L_{\alpha,\beta}$ there is a proof from $\text{ZFC}$ for $\sigma$ or $\neg \sigma$?

Definition 2: Let $T$ be a first order theory and $\sigma$ a sentence. Define the "distance of $\sigma$ from $T$" as follows: $d(T,\sigma

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!

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 replace the notion of first order logic ($L_{\omega,\omega}$) which uses finite proofs with some infinitary logic like $L_{\alpha,\beta}$ and look at the the theory $\text{ZFC}$ using this new glasses.

Definition 1: A theory $T$ in logic $L_{\alpha , \beta}$ is "totally consistent" if for all ordinals $\gamma\geq \alpha , \delta\geq \beta$ the theory $T$ is consistent in logic $L_{\gamma,\delta}$ too.

Question 1: Is the $L_{\omega , \omega}$ - theory $\text{ZFC}$ totally consistent? In the other words: is it possible to prove an inconsistency from $\text{ZFC}$ by allowing proofs in higher length in infinatry logics?

Question 2: Is any unprovable sentence from $\text{ZFC}$ somewhere decidable? Precisely: let $\sigma$ be an unprovable sentence from $\text{ZFC}$ in logic $L_{\omega ,\omega}$. Are there two large enough ordinals $\alpha , \beta$ such that in the logic $L_{\alpha,\beta}$ there is a proof from $\text{ZFC}$ for $\sigma$ or $\neg \sigma$?

Definition 2: Let $T$ be a first order theory and $\sigma$ a sentence. Define the "distance of $\sigma$ from $T$" as follows: $d(T,\sigma):=min\{\alpha\in Ord~|~\sigma~\text{is a theorem of}~T~\text{in logic}~L_{\alpha,\alpha}\}$ and if $\{\alpha\in Ord~|~\sigma~\text{is a theorem of}~T~\text{in logic}~L_{\alpha,\alpha}\}=\emptyset$ define $d(T,\sigma)=\infty$.

Remark 1: We have $d(\text{ZFC},2^{\aleph_{0}}\geq \aleph_1)\leq\omega$.

Question 3: What are the values of $d(\text{ZFC},\text{CH}),d(\text{ZFC},\neg \text{CH})$?

Remark 2: The following (imaginary) diagram says that $d(\text{ZFC},\text{CT})\leq\omega\leq d(\text{ZFC},\neg \text{CH})\leq\alpha<d(\text{ZFC},\text{CH})\leq\beta$ ($\text{CT}$ 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!