#it is possible to write any finite set of axioms as an isomorphism to axioms about the natural numbers
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Surprising absolutely noone except undergraduate math students, it turns out math actually is only about numbers in the end!
#math#mathblr#mathematics#oh so you thought abstract algebra stopped being about numbers to get to more complex structures#and you thought fundamental logic is so abstract exactly because it is more general than just numbers#well guess what#gödel wants to have a word with you#my friend every statement that can possibly be made by humans ever can be made isomorphic to some structure of the natural numbers#even if that structure is a function space of the power set of the power set of the natural numbers#it is possible to write any finite set of axioms as an isomorphism to axioms about the natural numbers#however since the axiomatic definition of the naturals gives you some axioms already it is important to note that for some really fucked up#axiomatic systems you have to explicitly include axioms that prevent you from using thise axioms in sone ways#as in you can't use the set theorethic definition to choose elements of a set for example#you have to choose elements from a set with choice functions constructible from axioms provided#now for making set theory equivalent to the naturals the choice functions you can derive are equivalent to just choosing a natural#but it could be not the case
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