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Suppose we have a system with a set of axioms. Now we begined to define new terms. E.g. in maths we have a particular set of axioms and then we define what a function is. But do all the functions automatically start to exist? For example, did the exponential function came into existence because we defined the term function? Or it was already there and we just give a name of what function is so we can say that exponential function is a member of the set of functions?

Example Consider now the statement "All functions are even". Where does the "all functions" refer? All the functions that exist? If no function exists then the term "all functions" fail to refer. So should we prove existence before we assign a truth value to that statement?

I have read also the post Do numbers exist independently from observers?

But I am asking why in a given system with axioms the the objects we assign properties are already exist so we have not the problem of "fail to refer".

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