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Let us define a set N1 to be the set of all natural numbers greater than 0.

We can define an enumeration function from the naturals to this set N1 like so:

Let us define another set which we shall call T (even though it can be easily seen that T = N), which we shall base on the union of our set N1 with the set containing only the number 0.

We can enumerate this function with a basic function

Since we have an enumeration function with an unbounded upper value for x for both N1 and T, both sets have the same cardinality of that of the natural numbers. However, surely this cannot be right as |N1 = |T - 1 (as N1 is the same as T less 1 element).

So we either have:

or

Wat.