It might be easier to think of it this way: Are there more natural numbers than there are <i>even</i> natural numbers?<p>The answer is no, as illustrated by the Hotel paradox[0], in which we have a infinitely many rooms and want to accommodate a (possibly infinite) number of guests.. To summarize: For any <i>finite</i> number of guests, you can always find an even-numbered room to correspond to that guest (assume the guests are numbered sequentially, then double their number and put them in the room that has that number.). This creates a one-to-one correspondence, which means that the sets are the same size.<p>You might say, &#x27;well, that only works because we&#x27;re dealing with a finite number of guests. But we&#x27;re talking about infinity&#x27;. There are a few different ways of answering that question. In my opinion, the easiest way to look at it is to remember that there <i>is</i> no such number as &#x27;infinity&#x27; - when we say &#x27;infinity&#x27;, we&#x27;re really trying to express the concept of growing without bound. So, the above strategy (double the person&#x27;s number) works for any arbitrarily large group. At no point does it <i>stop</i> working, even as the group size grows larger and larger, so we can say that the two sets have the same size.<p>On the other hand, we have no such strategy for putting every <i>irrational</i> number in one-to-one correspondence with the counting numbers. That proof is a little harder, and the analogy with the hotel guests breaks down, unfortunately, so it&#x27;s a bit tougher to explain.<p>[0] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Hilbert&#x27;s_paradox_of_the_Grand_Hotel" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Hilbert&#x27;s_paradox_of_the_Grand...</a>