Path: ...!weretis.net!feeder9.news.weretis.net!i2pn.org!i2pn2.org!.POSTED!not-for-mail From: Richard Damon Newsgroups: comp.theory Subject: Re: Cantor Diagonal Proof Date: Fri, 11 Apr 2025 09:24:57 -0400 Organization: i2pn2 (i2pn.org) Message-ID: <9539f14d4f49fd27da02d073e3306bafc01b7186@i2pn2.org> References: <4a8988895deed610068ddca65068842dbc20bf7d@i2pn2.org> <9c1620498b305f37e3f05840074d08af916377fc@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Fri, 11 Apr 2025 13:24:57 -0000 (UTC) Injection-Info: i2pn2.org; logging-data="4091536"; mail-complaints-to="usenet@i2pn2.org"; posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg"; User-Agent: Mozilla Thunderbird X-Spam-Checker-Version: SpamAssassin 4.0.0 In-Reply-To: Content-Language: en-US Bytes: 3590 Lines: 37 On 4/11/25 3:26 AM, Lawrence D'Oliveiro wrote: > On Thu, 10 Apr 2025 21:21:18 -0400, Richard Damon wrote: > >> On 4/10/25 8:28 PM, Lawrence D'Oliveiro wrote: >>> >>> On Wed, 9 Apr 2025 22:00:11 -0400, Richard Damon wrote: >>> >>>> But a finite list can't get you to the needed arbitrary precision >>>> needed. >>> >>> I was going to say, sure it can, because the size of the list is a >>> function of the precision you ask for. >> >> But the function needs to be prepared to handle ANY precision, and thus >> needs to be infinite. > > That’s true of computable numbers in general, so unless you’re objecting > to the very existence of the concept, it’s still irrelevant. > >> But you need to remember that he wasn't "constructing" it in the manner >> you are assuming, it isn't being constructed by a finite function, as >> that wasn't the domain he was talking about. > > Given the example list I gave elsewhere, there is a fundamental conflict > between a proof by induction (a well-established technique) and a proof by > his construction (which has to be seen as something novel). I would say > that points to a logical weakness in his construction. Remember, Cantor wasn't talking about "Computable" numbers, his list was a proported list of all reals mapped one to one with the Natural Numbers, something doable if they were countable. The problem with your "induction" is you assumed the existance of a computation that doesn't exist, a computation that given the nth digit of the nth computable number for any value of n. Such a construction, by necessity to work for ALL n, must have an infinite algorithm, and thus can't be just assumed to exist.