An intutive counterexample to the axiom of choice
Perhaps we should start admitting to ourselves that infinity is not really something that actually exists but instead it&#x27;s just an imaginary concept that we created but that has no correspondence to anything that makes sense in our universe?<p>Or at least, some versions of infinity (I will leave which ones, exactly, to the experts).<p>Hence all the paradoxes that derive from it.<p>I mean, perhaps we can work with <i>some</i> version(s) of infinity, without creating a plethora of paradoxes.<p>The natural numbers, for example, might be something that is infinite and makes sense. However, perhaps talking about a <i>single set</i> that contains an infinite amount of natural numbers might not make sense (or perhaps it still might?). Or perhaps we just need a new language for talking about infinite things without re-using the language we use for describing (finite) sets, so that we don&#x27;t try to reuse concepts from finite objects with infinite objects.<p>Or perhaps the only objects that exist (and can be meaningfully talked about) are the computable ones. And&#x2F;or perhaps &quot;computable function&quot; is one that can use infinite amounts of time but only finite amounts of storage (rather than infinite amounts of storage).<p>Would it make sense to go back to classical finitism and re-evaluate some of the choices that were made along the way to where we are now?<p>I can&#x27;t help but think about the many paradoxes that arise from the very (supposed) existence of infinity.<p>For example, everybody thinks the halting problem is undecidable. The halting problem is clearly not undecidable on finite-state machines (i.e. real computers), it&#x27;s only undecidable if you use imaginary Turing machines (which supposedly can have literally infinite state) as a model. But of course, Turing machines cannot exist in our universe.<p>So it&#x27;s rather ironic that people say the Halting problem is undecidable, when this is only true for (impossible to construct) imaginary machines but it&#x27;s not true for the real machines that we actually have and that are possible to construct.<p>Or think about Hilbert&#x27;s paradox of the Grand Hotel.<p>Or the Banach-Tarski paradox.<p>Or Galileo&#x27;s paradox.<p>Or other such paradoxes that only exist if you believe that imaginary infinite objects could possibly exist.<p>Most people will tell you &quot;well, it&#x27;s just that our intuition doesn&#x27;t work for infinite things&quot;, but in this specific instance, it&#x27;s really hard for me to believe that it&#x27;s our intuitions that are mistaken rather than that we actually made some conceptual mistakes along the way. To me, this argument sounds more like gaslighting rather than a real convincing argument.<p>I&#x27;m not saying our intuition is always correct, though. For example, the quantum properties of our universe are clearly not intuitive but it&#x27;s hard to argue that our universe is not quantum, after all the scientific experiments that have clearly verified this to be true.<p>I don&#x27;t think that invalidates my argument, though, because the convincing argument in that case is not &quot;sorry, you must believe my theory that the universe is quantum&quot; despite not existing any evidence that this was true, but rather, &quot;sorry, but here&#x27;s all this real evidence that our universe truly is quantum&quot;.<p>Am I crazy here?