<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>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></item><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>librexpr</author><text>I think if we take &amp;quot;description of a number&amp;quot; to mean &amp;quot;ZF formula that uniquely picks out that number&amp;quot;, then that cannot be defined, because a formula picks out a number when it is true for that number and false for all others, but by Tarski[0], the truth predicate cannot be defined inside the logic itself. So &amp;quot;the set of all numbers which cannot be described&amp;quot; cannot be talked about using ZF.&lt;p&gt;However, there is a way around it, by taking as axiom that ZF is consistent, choosing some model M of ZF, and then talking about the set S of numbers inside M that cannot be described.&lt;p&gt;[0] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Tarski%27s_undefinability_theorem&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Tarski%27s_undefinability_theo...&lt;/a&gt;</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>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></item><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>LegionMammal978</author><text>It&amp;#x27;s not too mind-boggling: we can describe the set as a whole without any issues, we just can&amp;#x27;t &lt;i&gt;uniquely&lt;/i&gt; describe any particular number in the set. That doesn&amp;#x27;t stop us from generalizing over all numbers in the set, or even over all numbers in a describable subset, such as &amp;quot;the set of all undescribable primes&amp;quot;.</text></comment>