[LudwigNagasena]

You can’t axiom your way out of 1 apple and 1 apple being 2 apples together. So axioms are not really that arbitrary.

[amavect]

You implicitly used an axiom to ignore the differences between the apples. Someone else could use different axioms to talk about the sizes of the apples (1 large + 1 small = ?), or the color of the apples (1 red + 1 green = ?), or the taste of the apples (1 sweet + 1 sour = ?).

People "axiom" their way out of 1+1=2 in this way: by changing the axioms, they change the topic, so they change the conclusion. I observe this pattern in disagreements very often.

[LudwigNagasena]

I have used appropriate axioms, not arbitrary axioms. If you want to talk about size or color or taste, you would use “axioms” appropriate for you case.

[scoofy]

They are, by definition. The reason why we choose them is exactly to map a deductive framework onto an inductive reality.

[LudwigNagasena]

That doesn’t seem to match the definition of “arbitrary”.

[scoofy]

We can choose whichever axioms we want. There is still arguments over the axiom of choice, but nobody cares because it’s entirely helpful.