The countable set of real numbers [video]
Stupid question form a ignorant person (me): Are reals actually real? Do they even exist?<p>Almost <i>all</i> reals (besides some countable subsets, which are irrelevant compared to the many many more reals that aren&#x27;t part of those sets) have infinite many non repeating decimal places, and what&#x27;s even more annoying, almost all of them aren&#x27;t computable.<p>The physical universe we can experience is finite. How can more than infinite many &quot;unspeakable&quot; things &quot;exist&quot; in that universe? How does this make sense? You can&#x27;t express most reals in <i>any</i> meaningful way! Not even by constructing imaginary &quot;infinite&quot; computers.<p>So, when almost all reals aren&#x27;t anything we could ever experience, or even name in any meaningful way, why do people believe they &quot;exist&quot; in the first place?<p>I have no issue to imagine infinite things as long as I can construct some algorithm to name them. But when there is no such algorithm, how can we know that the thing that can&#x27;t be named actually exists?<p>I&#x27;m not trolling. It&#x27;s a genuine question I have since some time but didn&#x27;t have the chance to talk to some mathematicians to get things explained. Maybe someone here could try to explain why we should believe in the existence of real numbers, even there is nothing in the whole universe that relates to most of them?<p>(Especially as the possibility to subdivide any interval infinitely gives rise to phenomenons like the Banach–Tarski paradox. But most people would agree: You can&#x27;t make something out of nothing. This is imho another strong pointer that reals are an illogical idea in the end. They give rise to contradictions to basic things we assume to know about reality).