Think you may want to read about what an axiom is.
An axiom, by the very definition, cannot be proven.
Since the mathematical impossibility of exactly once delivery can be proven (there exists a proof), it means it is not an axiom.
Now, a mathematical proof is different from physical reality. In mathematics 10^(10^(10^10)) is different from infinity while in reality it is not. In mathematics you can halve distance between you and a point every second and you will never reach it while in physical world you will reach it in less than a minute. In mathematics you can win any lottery however low your win chance is by just trying it an infinite number of times. In real world you can't.
An anecdote (funny regardless of whether true) says that CIA cryptographers created once an unbreakable encryption scheme using one time pad. One time pad is mathematically proven to be impossible to break.
USSR cryptographers promptly learned to decipher all cryptograms.
Apparently, when combining plaintext and one time pad the machine would generate different voltages depending on the bit used in the pad, so rather than output 0V for false and 1V for true, the output became 0.9V or 1.1V for true and 0V or 0.1V for false depending on what bit was used in the pad.
This shows how idealised world is different from engineering reality and forgetting about it can lead to large errors in judgment.
The same kind of errors in judgment as people saying "you cannot have exactly-once delivery".
Programs work in real world and not in idealised mathematical space. In real world, exactly once delivery is a solved problem for any practical purpose. Which is evidenced by all those systems that actually do, in fact, provide exactly once delivery.
An axiom, by the very definition, cannot be proven.
Since the mathematical impossibility of exactly once delivery can be proven (there exists a proof), it means it is not an axiom.
Now, a mathematical proof is different from physical reality. In mathematics 10^(10^(10^10)) is different from infinity while in reality it is not. In mathematics you can halve distance between you and a point every second and you will never reach it while in physical world you will reach it in less than a minute. In mathematics you can win any lottery however low your win chance is by just trying it an infinite number of times. In real world you can't.
An anecdote (funny regardless of whether true) says that CIA cryptographers created once an unbreakable encryption scheme using one time pad. One time pad is mathematically proven to be impossible to break.
USSR cryptographers promptly learned to decipher all cryptograms.
Apparently, when combining plaintext and one time pad the machine would generate different voltages depending on the bit used in the pad, so rather than output 0V for false and 1V for true, the output became 0.9V or 1.1V for true and 0V or 0.1V for false depending on what bit was used in the pad.
This shows how idealised world is different from engineering reality and forgetting about it can lead to large errors in judgment.
The same kind of errors in judgment as people saying "you cannot have exactly-once delivery".
Programs work in real world and not in idealised mathematical space. In real world, exactly once delivery is a solved problem for any practical purpose. Which is evidenced by all those systems that actually do, in fact, provide exactly once delivery.