Path: news.eternal-september.org!eternal-september.org!feeder3.eternal-september.org!news.szaf.org!news.karotte.org!news.space.net!news.muc.de!.POSTED.news.muc.de!not-for-mail From: Alan Mackenzie Newsgroups: sci.math Subject: Re: The non-existence of "dark numbers" Date: Thu, 13 Mar 2025 17:04:12 -0000 (UTC) Organization: muc.de e.V. Message-ID: References: <8734fghp5m.fsf@bsb.me.uk> Injection-Date: Thu, 13 Mar 2025 17:04:12 -0000 (UTC) Injection-Info: news.muc.de; posting-host="news.muc.de:2001:608:1000::2"; logging-data="41687"; mail-complaints-to="news-admin@muc.de" User-Agent: tin/2.6.4-20241224 ("Helmsdale") (FreeBSD/14.2-RELEASE-p1 (amd64)) Ben Bacarisse wrote: > Alan Mackenzie writes: >> WM wrote: >>> On 12.03.2025 18:42, Alan Mackenzie wrote: >>>> WM wrote: >>>>> If the numbers are definable. >>>> Meaningless. Or are you admitting that your "dark numbers" aren't >>>> natural numbers after all? >>> They >> They? >>>>> Learn what potential infinity is. >>>> I know what it is. It's an outmoded notion of infinity, popular in the >>>> 1880s, but which is entirely unneeded in modern mathematics. >>> That makes "modern mathematics" worthless. >> What do you know about modern mathematics? You may recall me challenging >> others in another recent thread to cite some mathematical result where >> the notion of potential/actual infinity made a difference. There came no >> coherent reply (just one from Ross Finlayson I couldn't make head nor >> tail of). Potential infinity isn't helpful and isn't needed anymore. > WMaths does (apparently) have one result that is not a theorem of modern > mathematics. In WMaths there sets P and E such that > E in P and P \ {E} = P > WM himself called this a "surprise" but unfortunately he has never been > able to offer a proof. Surprise indeed, but no surprise, too. He's coming pretty close to assertions like that in his replies to me, in the bits which are coherent. > On another occasion he and I came close to another when I defined a > sequence of rationals that he agreed was monotonic, increasing and > bounded above but which (apparently) does not converge to a real in > WMaths. It was "defined enough" to be monotonic and bounded but not > "defined enough" to converge to a real. Maybe it converged to some function of dark numbers. They're not real, are they? ;-) But he's clearly ignorant of the axiom of completion (the one which distinguishes the reals from ordered subfields). > But, in general, he hates talking about his WMaths because it gets him > into these sorts of blind alleys. But he loves going on about "dark numbers" and apparently still believes they exist. Hmmm. > -- > Ben. -- Alan Mackenzie (Nuremberg, Germany).