tag:blogger.com,1999:blog-5592542.post558708398447570486..comments2024-10-01T18:06:36.719-07:00Comments on Rondam Ramblings: An intutive counterexample to the axiom of choiceRonhttp://www.blogger.com/profile/11752242624438232184noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-5592542.post-56501347650742674142023-01-24T00:39:11.033-08:002023-01-24T00:39:11.033-08:00@Ron You have not actually proved that such a set,...@Ron You have not actually proved that such a set, containing all possible descriptions and all possible diagonalizations, exists within set theory. In fact, diagonalization shows that such a set is not definable in set theory!<br /><br />To put it differently: your counterpoint that my original description system is incomplete is not important: my description of y was given using English, which Guyhttps://www.blogger.com/profile/00193857548226023355noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-31614852355364582682023-01-23T19:28:30.651-08:002023-01-23T19:28:30.651-08:00@CientistaNãoAcadêmico:
> we do expand alphabe...@CientistaNãoAcadêmico:<br /><br />> we do expand alphabets all the time<br /><br />Sure, but they are nonetheless always bounded, which means that the number of possible descriptions is still countable.<br />Ronhttps://www.blogger.com/profile/11752242624438232184noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-42993732841523289332023-01-23T11:55:16.321-08:002023-01-23T11:55:16.321-08:00"Describable" depends on a choice of alp..."Describable" depends on a choice of alphabet (basically, any finite set is an alphabet and s symbol is just an element of an alphabet) and grammar for well-formed formulas on that alphabet, and thats completely arbitrary. Non-describable will be more of a limitation of my choice of language than a characteristic of the non-describable object.<br /><br />And we do expand alphabets all CientistaNãoAcadêmicohttps://www.blogger.com/profile/03444854452913702011noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-5462535876808163942023-01-23T10:36:39.420-08:002023-01-23T10:36:39.420-08:00[Corrected version]
@Guy:
> Clearly y is diff...[Corrected version]<br /><br />@Guy:<br /><br />> Clearly y is different from any of your describable reals! But I've just described it, so what gives?<br /><br />What gives is that your original description system "whatever system you like" was incomplete, as all such description systems necessarily are. But that doesn't matter to my argument, which includes *all possible* Ronhttps://www.blogger.com/profile/11752242624438232184noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-75792431684023569132023-01-23T07:55:38.931-08:002023-01-23T07:55:38.931-08:00Special irrationals like pi and e are described by...Special irrationals like <i>pi</i> and <i>e</i> are described by their specialness. Doesn't the diagonal argument for irrationals being a higher order of infinity depend on the Axiom of Choice? Maybe the problem is not with Choice but continuity.CCubedhttps://www.blogger.com/profile/09857086025354354185noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-76902825175425615902023-01-23T07:43:02.258-08:002023-01-23T07:43:02.258-08:00I don't think your example has anything to do ...I don't think your example has anything to do with the axiom of choice, which informally states that "given any collection of sets, you can pick an element from each of them". Set theory has no problem picking elements from a _single_ set, even without choice, provided you can prove that it is non-empty!<br /><br />And indeed, ordinary set theory (even without choice) proves that Guyhttps://www.blogger.com/profile/00193857548226023355noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-29304102474075452972023-01-23T07:18:37.582-08:002023-01-23T07:18:37.582-08:00@Unknown:
> symbols themselves are not really ...@Unknown:<br /><br />> symbols themselves are not really countable<br /><br />Symbols have to be drawn from a finite set. That is part of the standard definition of a symbol.<br /><br />@ Kaa1el:<br /><br />> I assume by "description" it means the "definable set" (https://en.wikipedia.org/wiki/Definable_set) or "definable number" (https://en.wikipedia.org/wikiRonhttps://www.blogger.com/profile/11752242624438232184noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-10458443807020757162023-01-23T05:37:43.199-08:002023-01-23T05:37:43.199-08:00Great start, but I think this misses the final ste...Great start, but I think this misses the final step. <br /><br />There’s no particular problem with claiming, in isolation, to have picked one element from some infinite collection of non-empty sets. The problem arises when you try to use that to construct an argument or proof.<br /><br />In each step where your argument depends on the existence of some set of unambiguous specific choices , then,Simonhttps://www.blogger.com/profile/14234378992627387956noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-31589957331377436002023-01-23T05:28:22.026-08:002023-01-23T05:28:22.026-08:00I assume by "description" it means the &...I assume by "description" it means the "definable set" (https://en.wikipedia.org/wiki/Definable_set) or "definable number" (https://en.wikipedia.org/wiki/Definable_real_number) in model theory using some finitary language like FOL. So they have explicit definition in mathematics. It is a countable set by definition, however it is only "countable" under the Kaa1elhttps://www.blogger.com/profile/16334652072773231421noreply@blogger.comtag:blogger.com,1999:blog-5592542.post-62904467597271724082023-01-23T05:03:31.083-08:002023-01-23T05:03:31.083-08:00I wonder if such a set really can exist.
First, a...I wonder if such a set really can exist.<br /><br />First, an element that cannot be described by using _any_ finite collection of symbols might not exist. The number of finite collections of symbols is not neccesarity countable (symbols themselves are not really countable: if a symbol is e.g. a trace of a pen on paper, it's a curve in 2D space).<br /><br />Second, if we can denote the set ofAnonymoushttps://www.blogger.com/profile/15450840687623781601noreply@blogger.com