<story><title>An intutive counterexample to the axiom of choice</title><url>http://blog.rongarret.info/2023/01/an-intutive-counterexample-to-axiom-of.html</url></story><parent_chain><item><author>boole1854</author><text>The argument here seems to rely on a natural language intuition, or perhaps even a physical or computational intuition, about what it mean to &amp;quot;choose&amp;quot; something from a collection.&lt;p&gt;However, remember that the axiom of choice is technically a statement about the &lt;i&gt;existence of certain functions&lt;/i&gt;, specifically functions that map from any set of non-empty sets to elements of those sets. Whether or not the existence of those functions says something about the possibility of &amp;quot;choosing&amp;quot; is more of a terminological question about how to connect informal natural language to mathematics than it is a substantive mathematical question.&lt;p&gt;Turning to the mathematical question itself, it seems to me that the author&amp;#x27;s intuitions fare poorly. The author argues that we apparently cannot &amp;quot;choose&amp;quot; an element from a set that contains only numbers that are not finitely describable. But the mathematical question is about the existence of a mapping function, and it seems that whether or not the numbers in each set are finitely describable is irrelevant to this question:&lt;p&gt;Consider the function &lt;i&gt;f(x) = x + 1&lt;/i&gt;. This function is defined for all real numbers &lt;i&gt;x&lt;/i&gt;. It maps each real number to another real number. Note that the domain of the function includes even those real numbers that are not finitely describable. The meaning of the function itself is nevertheless clear and the function is easy to write down. The fact that the domain of the function includes indescribable numbers is not relevant to the question of whether such a function exists mathematically.</text></item></parent_chain><comment><author>ketzu</author><text>I wonder if the set of undescribable numbers is well defined at all. It seems to run into the same problem as the &amp;quot;minimum un-interesting number is interesting&amp;quot; as a function chosing a number from that set would be a finite description of that number, removing it from the set. (This already happens if you could select a finite subset of those numbers, as finite numbers could be ordered and you could just select the minimum.)&lt;p&gt;edit: Hm... thinking about it. The finitely-many-symbols argument also only holds if you consider finitely many interpretations of those symbols and if there actually are only finitely many symbols.</text></comment>
<story><title>An intutive counterexample to the axiom of choice</title><url>http://blog.rongarret.info/2023/01/an-intutive-counterexample-to-axiom-of.html</url></story><parent_chain><item><author>boole1854</author><text>The argument here seems to rely on a natural language intuition, or perhaps even a physical or computational intuition, about what it mean to &amp;quot;choose&amp;quot; something from a collection.&lt;p&gt;However, remember that the axiom of choice is technically a statement about the &lt;i&gt;existence of certain functions&lt;/i&gt;, specifically functions that map from any set of non-empty sets to elements of those sets. Whether or not the existence of those functions says something about the possibility of &amp;quot;choosing&amp;quot; is more of a terminological question about how to connect informal natural language to mathematics than it is a substantive mathematical question.&lt;p&gt;Turning to the mathematical question itself, it seems to me that the author&amp;#x27;s intuitions fare poorly. The author argues that we apparently cannot &amp;quot;choose&amp;quot; an element from a set that contains only numbers that are not finitely describable. But the mathematical question is about the existence of a mapping function, and it seems that whether or not the numbers in each set are finitely describable is irrelevant to this question:&lt;p&gt;Consider the function &lt;i&gt;f(x) = x + 1&lt;/i&gt;. This function is defined for all real numbers &lt;i&gt;x&lt;/i&gt;. It maps each real number to another real number. Note that the domain of the function includes even those real numbers that are not finitely describable. The meaning of the function itself is nevertheless clear and the function is easy to write down. The fact that the domain of the function includes indescribable numbers is not relevant to the question of whether such a function exists mathematically.</text></item></parent_chain><comment><author>kybernetikos</author><text>&amp;gt; But the mathematical question is about the existence of a mapping function, and it seems that whether or not the numbers in each set are finitely describable is irrelevant to this question:&lt;p&gt;If it&amp;#x27;s possible to construct the set at all, then there is by definition no function that can be described in a finite number of symbols that can select an element from the set (since if a finitely describable function can define a member of the set, it shouldn&amp;#x27;t be in the set - you&amp;#x27;d define the element as being the result of the function).&lt;p&gt;I don&amp;#x27;t know what the scholarly view is on relying on functions that &lt;i&gt;cannot be defined in any human language, including mathematics&lt;/i&gt;.  I would intuitively lean towards not allowing them in proofs, but perhaps they&amp;#x27;re useful mathematically?&lt;p&gt;Another interesting aspect - these are numbers, so presumably there is a smallest one, but identifying that is definitionally ruled out too, since &amp;#x27;smallest element in the set of numbers indescribable in a finite sequence of symbols&amp;#x27;, is itself a finite sequence of symbols.</text></comment>