Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).

Their non-existence is consistent with axioms of usual mathematics.

It is provable that some of them don't exist at all.

They show many unusual strange properties.

$\vdots$

These are a part of arguments which could be used against large cardinal axioms but not only many set theorists believe on existence of large cardinals but also they refute every statement like $V=L$ which is contradictory with their existence.

What makes large cardinal axioms reasonable enough to add them to set of axioms of usual mathematics? Is there any particular mathematical or philosophical reason which forces/convinces us to accept large cardinal axioms? Is there any fundamental axiom which is philosophically reasonable and implies the necessity of adding large cardinals to mathematics? Is it Reflection Principle which informally says "all properties of the universe $V$ should be reflected in a level of von Neumann's hierarchy" and so because within $\text{ZF-Inf}$ the universe $V$ is infinite we should add large cardinal $\omega$ (which is inaccessible from finite numbers) by accepting large cardinal axiom $\text{Inf}$ and because $V$ is a model of (second order) $\text{ZFC}$ we should accept existence of inaccessible cardinals to reflect this property to $V_{\kappa}$ for $\kappa$ inaccessible and so on?

Question. I am searching for useful mathematical, philosophical,... references which investigate around possible answers of above questions.

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).

Their non-existence is consistent with axioms of usual mathematics.

It is provable that some of them don't exist at all.

They show many unusual strange properties.

$\vdots$

These are a part of arguments which could be used against large cardinal axioms, but many set theorists not only believe in the existence of large cardinals, but also refute every statement like $V=L$ which is contradictory to their existence.

What makes large cardinal axioms reasonable enough to add them to set of axioms of usual mathematics? Is there any particular mathematical or philosophical reason which forces/convinces us to accept large cardinal axioms? Is there any fundamental axiom which is philosophically reasonable and implies the necessity of adding large cardinals to mathematics? Is it Reflection Principle which informally says "all properties of the universe $V$ should be reflected in a level of von Neumann's hierarchy" and so because within $\text{ZF-Inf}$ the universe $V$ is infinite we should add large cardinal $\omega$ (which is inaccessible from finite numbers) by accepting large cardinal axiom $\text{Inf}$ and because $V$ is a model of (second order) $\text{ZFC}$ we should accept existence of inaccessible cardinals to reflect this property to $V_{\kappa}$ for $\kappa$ inaccessible and so on?

Question. I am searching for useful mathematical, philosophical,... references which investigate around possible answers of above questions.

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).

Their non-existence is consistent with axioms of usual mathematics.

It is provable that some of them don't exist at all.

They show many unusual strange properties.

$\vdots$

These are a part of arguments which could be used against large cardinal axioms but not only many set theorists believe on existence of large cardinals but also they refute every statement like $V=L$ which is contradictory with their existence.

What makes large cardinal axioms reasonable enough to add them to set of axioms of usual mathematics? Is there any particular mathematical or philosophical reason which forces/convinces us to accept large cardinal axioms? Is there any fundamental axiom which is philosophically reasonable and implies the necessity of adding large cardinals to mathematics? Is it Reflection Principle which informally says "all properties of the universe $V$ should be reflected in a level of von Neumann's hierarchy" and so because within $\text{ZF-Inf}$ the universe $V$ is infinite we should add large cardinal $\omega$ by accepting large cardinal axiom $\text{Inf}$ and because $V$ is a model of (second order) $\text{ZFC}$ we should accept existence of inaccessible cardinals to reflect this property to $V_{\kappa}$ for $\kappa$ inaccessible and so on?

Question. I am searching for useful mathematical, philosophical,... references which investigate around possible answers of above questions.

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).

Their non-existence is consistent with axioms of usual mathematics.

It is provable that some of them don't exist at all.

They show many unusual strange properties.

$\vdots$

These are a part of arguments which could be used against large cardinal axioms but not only many set theorists believe on existence of large cardinals but also they refute every statement like $V=L$ which is contradictory with their existence.

What makes large cardinal axioms reasonable enough to add them to set of axioms of usual mathematics? Is there any particular mathematical or philosophical reason which forces/convinces us to accept large cardinal axioms? Is there any fundamental axiom which is philosophically reasonable and implies the necessity of adding large cardinals to mathematics? Is it Reflection Principle which informally says "all properties of the universe $V$ should be reflected in a level of von Neumann's hierarchy" and so because within $\text{ZF-Inf}$ the universe $V$ is infinite we should add large cardinal $\omega$ (which is inaccessible from finite numbers) by accepting large cardinal axiom $\text{Inf}$ and because $V$ is a model of (second order) $\text{ZFC}$ we should accept existence of inaccessible cardinals to reflect this property to $V_{\kappa}$ for $\kappa$ inaccessible and so on?

Question. I am searching for useful mathematical, philosophical,... references which investigate around possible answers of above questions.