What you are encountering is essentially the phenomenon first explicated by Godel; it is no less counterintuitive or phenomenal to me that $$ZF+\neg Con(ZF),$$ $$KP+\neg Con(KP),$$ $${\sf Homotopy\ Type\ Theory}+\neg Con({\sf Homotopy\ Type\ Theory}),$$ etc. are all consistent theories, but they are because consistency is a fundamentally metatheoretic concept, in the most inextricable sense of the word. This is highlighted in a quote from the text Inexhaustibility: A Non-Exhaustive Treatment by Torkel Franzén, who is himself quoting Gödel:

It is this theorem [the second incompleteness theorem] which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If somebody makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms.

That this insight is underivable from our axiomatic system of choice is an essential feature of any foundational system falling prey to the mild conditions in Gödel’s theorems, and because of this it makes perfect sense that we can assume our base theory to be inconsistent and have a new, consistent theory — consistency of our base theory was never provable to begin with, so adding the negation of this statement (or any other statement underivable from our axioms) will again yield a consistent system of axioms.

That this runs counter to our naïve intuition is understandable, but one of the main gifts offered forth by Gödel is this correction of basic human intuition; the sooner one adopts it as their new intuition, the sooner this apparently confusing and counterintuitive landscape of theories and consistency claims begins to take form.

What you are encountering is essentially the phenomenon first explicated by Gödel; it is no less counterintuitive or phenomenal to me that \begin{gather*} \newcommand\ZF{\mathsf{ZF}}\DeclareMathOperator\Con{\mathsf{Con}}\newcommand\KP{\mathsf{KP}} \ZF+\neg \Con(\ZF), \\ \KP+\neg \Con(\KP), \\ \textsf{Homotopy Type Theory}+\neg \Con(\textsf{Homotopy Type Theory}), \end{gather*} etc. are all consistent theories, but they are because consistency is a fundamentally metatheoretic concept, in the most inextricable sense of the word. This is highlighted in a quote from the text Inexhaustibility: A Non-Exhaustive Treatment by Torkel Franzén, who is himself quoting Gödel:

It is this theorem [the second incompleteness theorem] which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If somebody makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms.

That this insight is underivable from our axiomatic system of choice is an essential feature of any foundational system falling prey to the mild conditions in Gödel’s theorems, and because of this it makes perfect sense that we can assume our base theory to be inconsistent and have a new, consistent theory — consistency of our base theory was never provable to begin with, so adding the negation of this statement (or any other statement underivable from our axioms) will again yield a consistent system of axioms.

That this runs counter to our naïve intuition is understandable, but one of the main gifts offered forth by Gödel is this correction of basic human intuition; the sooner one adopts it as their new intuition, the sooner this apparently confusing and counterintuitive landscape of theories and consistency claims begins to take form.

What you are encountering is essentially the phenomenon first explicated by Godel; it is no less counterintuitive or phenomenal to me that $$ZF+\neg Con(ZF),$$ $$KP+\neg Con(KP),$$ $${\sf Homotopy\ Type\ Theory}+\neg Con({\sf Homotopy\ Type\ Theory}),$$ etc. are all consistent theories, but they are because consistency is (in some sense) a fundamentally metatheoretic concept, in the most inextricable sense of the word. This is highlighted in a quote from the text Inexhaustibility: A Non-Exhaustive Treatment by Torkel Franzén, who is himself quoting Gödel:

It is this theorem [the second incompleteness theorem] which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If somebody makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms.

That this insight is underivable from our axiomatic system of choice is an essential feature of any foundational system falling prey to the mild conditions in Gödel’s theorems, and because of this it makes perfect sense that we can assume our base theory to be inconsistent and have a new, consistent theory — consistency of our base theory was never provable to begin with, so adding the negation of this statement (or any other statement underivable from our axioms) will again yield a consistent system of axioms.

That this runs counter to our naïve intuition is understandable, but one of the main gifts offered forth by Gödel is this correction of basic human intuition; the sooner one adopts it as their new intuition, the sooner this apparently confusing and counterintuitive landscape of theories and consistency claims begins to take form.

What you are encountering is essentially the phenomenon first explicated by Godel; it is no less counterintuitive or phenomenal to me that $$ZF+\neg Con(ZF),$$ $$KP+\neg Con(KP),$$ $${\sf Homotopy\ Type\ Theory}+\neg Con({\sf Homotopy\ Type\ Theory}),$$ etc. are all consistent theories, but they are because consistency is a fundamentally metatheoretic concept, in the most inextricable sense of the word. This is highlighted in a quote from the text Inexhaustibility: A Non-Exhaustive Treatment by Torkel Franzén, who is himself quoting Gödel:

It is this theorem [the second incompleteness theorem] which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If somebody makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms.

That this insight is underivable from our axiomatic system of choice is an essential feature of any foundational system falling prey to the mild conditions in Gödel’s theorems, and because of this it makes perfect sense that we can assume our base theory to be inconsistent and have a new, consistent theory — consistency of our base theory was never provable to begin with, so adding the negation of this statement (or any other statement underivable from our axioms) will again yield a consistent system of axioms.

That this runs counter to our naïve intuition is understandable, but one of the main gifts offered forth by Gödel is this correction of basic human intuition; the sooner one adopts it as their new intuition, the sooner this apparently confusing and counterintuitive landscape of theories and consistency claims begins to take form.