How can some infinities be bigger than others?
I have no problem with the concept of cardinality of infinite sets. If you want to talk about the ability to map infinite sets, so that the cardinality of integers and even integers is the same, then great.<p>What drives me <i>bananas</i> is when anybody starts using the words &quot;size&quot; or &quot;larger&quot; or &quot;smaller&quot;. I will insist to my dying day that while the cardinality of even integers is the same as that of integers, the set of even integers is still <i>smaller</i> than the set of integers. That&#x27;s it&#x27;s still half the size.<p>After all, simply statistically, if I start sampling items randomly from the set of integers, I&#x27;ll quickly discover that it converges to half of them belonging to the set of even numbers, and half don&#x27;t.<p>And yes I <i>know</i> there are supposed theoretical problems with random sampling from an infinite set but honestly I don&#x27;t care. Pick <i>any</i> large bound you want from the set of integers, whether it&#x27;s from 1 million to 10 million, or negative a trillion to positive a trillion trillion trillion. It&#x27;s <i>always</i> going to converge to integers being twice the size of even integers.<p>I mean if we can deal with ratios in calculus down to infinitesimal sizes using limits, we can sure as heck go the other way, the limit as the bounds go to infinity <i>and the proportion still continues to hold perfectly</i>.<p>Somehow, at some point mathematicians just started treating cardinality as the <i>size</i> of sets, against all common sense, and you come across statements like &quot;the number of rational numbers is equal to the number of integers&quot;. Nonsense. They have the same <i>cardinality</i>, but they are <i>definitely</i> not the same <i>size</i>. It&#x27;s simply mathematical gaslighting and yes, I will die on this hill! :)