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It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to an individuals intuition and philosophical preferences.

Has there been any work done exploring what happens if we negate 'essential' axioms?

For example, what about $ZF$ with the axiom of extensionality negated?

My vague understanding of type theories is that axioms do not play such a central role, replaced largely by rules of inference (which are essentially axioms about reasoning instead of the objects of the theory, as far as I can tell? [*cue dunce cap]). But even in this setting, I think it is a coherent question to ask 'what if we take the negation of one of the 'essential' rules of inference as the rule instead?'.

This question is asked mostly as a lark; I am curious what the mathematical universe looks like when we explore a version of it with core pieces of our intuition 'reversed'. Any pointers are appreciated.

In my original edit, I claimed that just negating outright for universal axioms 'wouldn't have any interesting consequences' (which is wrong, see Noah's answer), and also that we might instead negate something like extensionality by claiming that equal sets never have the same elements, which is always wrong (see Wojowu and Naïm's comments).

It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to an individuals intuition and philosophical preferences.

Has there been any work done exploring what happens if we negate 'essential' axioms?

For example, what about $ZF$ with the axiom of extensionality negated?

My vague understanding of type theories is that axioms do not play such a central role, replaced largely by rules of inference (which are essentially axioms about reasoning instead of the objects of the theory, as far as I can tell? [*cue dunce cap]). But even in this setting, I think it is a coherent question to ask 'what if we take the negation of one of the 'essential' rules of inference as the rule instead?'.

This question is asked mostly as a lark; I am curious what the mathematical universe looks like when we explore a version of it with core pieces of our intuition 'reversed'. Any pointers are appreciated.

In my original edit, I claimed that just negating outright for universal axioms 'wouldn't have any interesting consequences' (which is wrong, see Noah's answer), and also that we might instead negate something like extensionality by claiming that equal sets never have the same elements, which is always wrong (see Wojowu's and Naïm's comments).

It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to an individuals intuition and philosophical preferences.

Has there been any work done exploring what happens if we negate 'essential' axioms?

For example, what about $ZF$ with the axiom of extensionality negated?

My vague understanding of type theories is that axioms do not play such a central role, replaced largely by rules of inference (which are essentially axioms about reasoning instead of the objects of the theory, as far as I can tell? [*cue dunce cap]). But even in this setting, I think it is a coherent question to ask 'what if we take the negation of one of the 'essential' rules of inference as the rule instead?'.

This question is asked mostly as a lark; I am curious what the mathematical universe looks like when we explore a version of it with core pieces of our intuition 'reversed'. Any pointers are appreciated.

A few minutes after posting, I realized that axioms with a universal flavor (e.g. choice) don't have interesting negations when we interpret 'negation' to literally mean 'prepend a $\neg$'; for instances like this, by 'negation' I would be satisfied with an axiom that expressed an intuition contrasting with the one captured by the usual axiom. So, for example, in the extensionality case the negation might not be 'there exist two sets with different elements which are equal' but rather that 'equal sets never have the same elements'. This is obviously impossible for finite sets, so we have to add some condition about the size of the sets, and this is where the forest begins to get dense enough that I hope someone else might have laid out some paths.

It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to an individuals intuition and philosophical preferences.

Has there been any work done exploring what happens if we negate 'essential' axioms?

For example, what about $ZF$ with the axiom of extensionality negated?

My vague understanding of type theories is that axioms do not play such a central role, replaced largely by rules of inference (which are essentially axioms about reasoning instead of the objects of the theory, as far as I can tell? [*cue dunce cap]). But even in this setting, I think it is a coherent question to ask 'what if we take the negation of one of the 'essential' rules of inference as the rule instead?'.

This question is asked mostly as a lark; I am curious what the mathematical universe looks like when we explore a version of it with core pieces of our intuition 'reversed'. Any pointers are appreciated.

In my original edit, I claimed that just negating outright for universal axioms 'wouldn't have any interesting consequences' (which is wrong, see Noah's answer), and also that we might instead negate something like extensionality by claiming that equal sets never have the same elements, which is always wrong (see Wojowu and Naïm's comments).

It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to an individuals intuition and philosophical preferences.

Has there been any work done exploring what happens if we negate 'essential' axioms?

For example, what about $ZF$ with the axiom of extensionality negated?

My vague understanding of type theories is that axioms do not play such a central role, replaced largely by rules of inference (which are essentially axioms about reasoning instead of the objects of the theory, as far as I can tell? [*queue dunce cap]). But even in this setting, I think it is a coherent question to ask 'what if we take the negation of one of the 'essential' rules of inference as the rule instead?'.

This question is asked mostly as a lark; I am curious what the mathematical universe looks like when we explore a version of it with core pieces of our intuition 'reversed'. Any pointers are appreciated.

A few minutes after posting, I realized that axioms with a universal flavor (e.g. choice) don't have interesting negations when we interpret 'negation' to literally mean 'prepend a $\neg$'; for instances like this, by 'negation' I would be satisfied with an axiom that expressed an intuition contrasting with the one captured by the usual axiom. So, for example, in the extensionality case the negation might not be 'there exist two sets with different elements which are equal' but rather that 'equal sets never have the same elements'. This is obviously impossible for finite sets, so we have to add some condition about the size of the sets, and this is where the forest begins to get dense enough that I hope someone else might have laid out some paths.

It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to an individuals intuition and philosophical preferences.

Has there been any work done exploring what happens if we negate 'essential' axioms?

For example, what about $ZF$ with the axiom of extensionality negated?

My vague understanding of type theories is that axioms do not play such a central role, replaced largely by rules of inference (which are essentially axioms about reasoning instead of the objects of the theory, as far as I can tell? [*cue dunce cap]). But even in this setting, I think it is a coherent question to ask 'what if we take the negation of one of the 'essential' rules of inference as the rule instead?'.

This question is asked mostly as a lark; I am curious what the mathematical universe looks like when we explore a version of it with core pieces of our intuition 'reversed'. Any pointers are appreciated.

A few minutes after posting, I realized that axioms with a universal flavor (e.g. choice) don't have interesting negations when we interpret 'negation' to literally mean 'prepend a $\neg$'; for instances like this, by 'negation' I would be satisfied with an axiom that expressed an intuition contrasting with the one captured by the usual axiom. So, for example, in the extensionality case the negation might not be 'there exist two sets with different elements which are equal' but rather that 'equal sets never have the same elements'. This is obviously impossible for finite sets, so we have to add some condition about the size of the sets, and this is where the forest begins to get dense enough that I hope someone else might have laid out some paths.