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My statement:

The first transfinite ordinal, omega is the first number that cannot be expressed by any natural number, therefore it is not included in the set of natural numbers. The set of natural numbers is a subset of real numbers, every natural number can be taken out of it, but still true that there is no integer number in it that is capable ("big enough") to pair with omega. By this, it is also a good statement to say that the set of transfinite ordinals and the real numbers are disjoint sets.

Is this a good tought? If this is not, then can be refined to make it mean that "omega is after the finite numbers therefore it is after also any real number"?

Please help me to make this statement more formal.

Is the Venn diagram below correct?

Thank you!

The first transfinite ordinal, omega is the first number that cannot be expressed by any natural number, therefore it is not included in the set of natural numbers. The set of natural numbers is a subset of real numbers, every natural number can be taken out of it, but still true that there is no integer number in it that is capable ("big enough") to pair with omega. By this, it is also a good statement to say that the set of transfinite ordinals and the real numbers are disjoint sets.

Is this a good tought? If this is not, then can be refined to make it mean that "omega is after the finite numbers therefore it is after also any real number"?

Please help me to make this statement more formal.

Is the Venn diagram below correct?

Thank you!