Recursive Definition Where ∞ is an Element of a Subset of Integers. How can this be true?

Question

I was recently provided the following True or False question:

Let S be a subset of the set of integers recursively defined by:
                Basis Step: 1 ∈ S
                Recursive Step: If x ∈ S, then x + 1 ∈ S.

           

True or False: ∞ ∈ S

I say false; however, I am being told by my professor it is true because,

"The basis step begins at 1 and the recursive step simply adds 1. The result of that addition is an element of set S. Then, 1 is added to that result and the corresponding result is also an element of S. This can go on all the way to infinity. This recursive definition is essentially for the set of all positive integers."

What I don't understand is how infinity can be an element of a subset of integers given that infinity is not an integer. . .

Answer 1

With the usual meaning attached to the symbol $\infty$, the claim is obviously false because

$$\infty\notin\mathbb Z$$ so that $$\infty\notin S\subset\mathbb Z.$$

With what you write in the title, the claim is obviously true because "$\infty$ is an Element of a Subset of Integers" is another way to write

$$\infty\in S.$$

In both cases, the recursive definition of $S$ is strictly of no use.