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

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. . .

• Your professor is wrong.If he means that the set includes infinitely many numbers, that would be true, but $\infty \in S$ is false. Jul 17, 2017 at 23:39
• As written, your professor is Not Even Wrong; $\infty$ (as such) does not have a standard 'standalone' definition. There are many ways of defining an item called $\infty$, but for none of them is this statement true by definition. Are you sure this is exactly what your professor wrote? Jul 17, 2017 at 23:42
• The claim is neither true nor false as long as $\infty$ has not been formally defined.
– user65203
Jul 17, 2017 at 23:42
• One wonders how someone with such little understanding of mathematical induction could be employed as a mathematics professor. Jul 17, 2017 at 23:58
• Every positive integer is in this set, and this set is infinite, but every member of the set is a finite number; there is nothing called $\infty$ that belongs to this set unless $\infty$ is some finite number. The professor is quite wrong. Jul 18, 2017 at 3:37

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.

• I wrote the title based on the problem at hand... the title is of my own making and wasn't something provided by the professor Jul 17, 2017 at 23:55
• @AntonRasmussen: I suspected this. Anyway, what you wrote is lacking rigor and has no truth value.
– user65203
Jul 17, 2017 at 23:57
• Cool, I'm gonna send a "can I get my points back?" email. I wanted to be wrong on this because this would be the fifth(!) email I've had to write pointing out mathematical errors that make the claimed correct answer to a quiz actually incorrect. He's given all points back so far ... still, I miss taking math classes under the math department and not the CS department! Jul 17, 2017 at 23:59
• @AntonRasmussen: as the claim is neither false nor true, he must give back the points and keep them.
– user65203
Jul 18, 2017 at 0:01
• Or, you know, throw out the question entirely? Thanks for your help on this by the way. Jul 18, 2017 at 0:10