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## Main Question or Discussion Point

In the past, I have asked in this forum about the concept of set membership, in the context of ZFC.

I guess it is a normal reaction to be a bit surprised by the usual statement in books that the set membership relationship is "undefined".

But I have had this idea: a typical definition of the natural numbers, by von Neumann, is that the number zero is defined as the empty set, the number one is defined as the set consisting of the empty set, and so on.

Since, at least from the Physics point of view, the only sets that we need are the numbers (and it is not problem to define, after the natural numbers, the integers, the rationals, the reals, the complexes, and then other mathematical objects out of those), we could proceed as follows:

We take ZFC at face value, without defining what set membership is. Set membership is just a binary relationship. ZFC axioms just tells us how objects, sets and the set membership relate to each other. We do not need to give a specific model of sets.

But even without defining what set membership is, we can define the number zero as the empty set, one as ... as above. And then, once we have the natural numbers, we define the integers ... and so on. So, we do not need to define what set membership is, in order to define and create the usual objects we use in mathematics. We only need to find the right properties, and define the right axioms (ZFC).

Is this argument correct?

I guess it is a normal reaction to be a bit surprised by the usual statement in books that the set membership relationship is "undefined".

But I have had this idea: a typical definition of the natural numbers, by von Neumann, is that the number zero is defined as the empty set, the number one is defined as the set consisting of the empty set, and so on.

Since, at least from the Physics point of view, the only sets that we need are the numbers (and it is not problem to define, after the natural numbers, the integers, the rationals, the reals, the complexes, and then other mathematical objects out of those), we could proceed as follows:

We take ZFC at face value, without defining what set membership is. Set membership is just a binary relationship. ZFC axioms just tells us how objects, sets and the set membership relate to each other. We do not need to give a specific model of sets.

But even without defining what set membership is, we can define the number zero as the empty set, one as ... as above. And then, once we have the natural numbers, we define the integers ... and so on. So, we do not need to define what set membership is, in order to define and create the usual objects we use in mathematics. We only need to find the right properties, and define the right axioms (ZFC).

Is this argument correct?