What is the largest ordinal number that is also a natural number?

Question

I know that $\omega$ is the first ordinal number that is not a natural number, so I was wondering what is the largest ordinal number that is also a natural number. I was guessing $\omega-1$, but since subtraction is undefined in ordinal (and cardinal) arithmetic, I ruled it out. I am quite sure that the answer to my question is equivalent to this supremum: $$\sup(\mathrm{On}\hspace{4px}\cap\mathbb{N})$$ Any help would be greatly appreciated.

Answer 1

Since every natural number is also an ordinal number, your question reduces to

"What is the largest natural number?"

Perhaps you already know the answer to this because, if $n$ is any candidate to be the "the largest natural number", then $n+1$ will beat it. That is, there is no such thing as the largest natural number.

You state "I am quite sure that the answer to my question is equivalent to this supremum: $\sup(\mathrm{On}\cap\mathbb N)$". The set $\mathrm{On}\cap\mathbb N$ is simply $\mathbb N$, and so $\sup(\mathrm{On}\cap\mathbb N)=\sup\mathbb N=\omega\notin\mathbb N$. The supremum of a set of ordinal numbers is not generally an element of that set. In the case when a set has a largest element, it is by definition an element of and the supremum of that set.