Why Some Infinities Are Bigger Than Others
The idea of &quot;differently sized infinities&quot; has always felt like a mathematical game to me.<p>The work of Cantor and beyond regarding infinite sets is generally taught as mathematical &quot;fact&quot;, as fundamental as 2+2=4, but unlike 2+2=4 is there any practical constraints or applications around it? It seems more like an almost arbitrary puzzle than &quot;fact&quot;.<p>I don&#x27;t disagree with any of the proofs or reasoning... it just kinda bugs me that the way the formal cardinality of infinite sets is communicated is as the commonsense notion of &quot;size&quot;.<p>When we make statements such as the size of the set of all natural numbers 1, 2, 3... is the same as the size of the set of all natural even numbers 2, 4, 6..., despite the former containing the latter but not vice-versa... it seems the word &quot;size&quot; -- and associated terminology &quot;larger than&quot;, &quot;smaller than&quot;, etc. -- is a particularly unhelpful set of words to have chosen for this.<p>The reason I think this is important is because if you go to e.g. the Wikipedia page for &quot;irrational number&quot;, it states in the introduction [1]:<p>&gt; <i>&quot;As a consequence of Cantor&#x27;s proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.&quot;</i><p>But it seems like an unwarranted leap to go from this formal comparison of cardinality of infinite sets, to the intuitive English-sentence idea that &quot;almost all&quot; real numbers are irrational, as if it had any analogy whatsover to a statement such as e.g. &quot;almost all&quot; values of 1&#x2F;x are defined.<p>Am I missing anything here? I&#x27;ve never heard of any natural practical application of this concept, in a way that suggests it&#x27;s &quot;true&quot; in the same way algebra and calculus are considered to be.<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Irrational_number" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Irrational_number</a>