Dieuodene's celebrated treatise in first volume only restricts to this axiom.
In Mathematics one only needs sets derived from N.
for example Euclids geomtry one can define a point as an ordered pair of real numbers.
So Mathematics does not need abstract sets and sets derived from N which is turn derived from the empty set will work.
So one can use restricted ZF theory where sets means those sets derived from N by constructions in ZF.
So we have every infinite set contains a countable set.
If one reads Marsdens analysis book the last two axioms of sewt theory are seldom needed in everyday Math. these are axiom of choice and the axiom which says that every mapping has a target
we do need cardinal and ordinal numbers and their arithmetic.
we do not need their definitions .
We just need the facts that Q has bijection with N, there can not exist bijection between R and N, and there can not exist a surjective map from P(X) to X.
So my question should be viewed in reference to these facts.
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