Path: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Thu, 02 Jan 2025 19:07:43 +0000 Subject: Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) Newsgroups: sci.logic,sci.math References: <881fc1a1-2e55-4f13-8beb-94d1f941b5af@att.net> <85194aeb-1b24-4486-8bcc-4dcd43b4fd2f@att.net> <2e188e21-4128-4c76-ba5d-473528262931@att.net> <3df3c8f4-05b1-477b-8812-f49bd46fa764@att.net> <672c2c2a-2f01-4cc6-9e2d-52c0f4bb2996@att.net> From: Ross Finlayson Date: Thu, 2 Jan 2025 11:07:43 -0800 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: <672c2c2a-2f01-4cc6-9e2d-52c0f4bb2996@att.net> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 94 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-eB4NBawDbxbevUT8G5u2xuXcugQJ3mrLcilrbIxaMZmlCLX+H2QNawyNPAXwILB0Sq5DQrqOcyx7Se2!nlHWE2Llh3bwRIk3NgO+TXTjzfLEUiXF+NfP+KjAoaZaCbR4yHzkZ09Q/v1iq5MoroyZaZXr5mU= X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 4838 On 01/02/2025 09:35 AM, Jim Burns wrote: > On 1/1/2025 6:50 PM, Ross Finlayson wrote: >> On 01/01/2025 01:14 PM, Jim Burns wrote: >>>> On 07/29/2024 12:46 PM, Ross Finlayson wrote: > >>>>> Or, you know, "infinity plus one". >>> >>> Consider the definition of a finite.cardinal as >>> the cardinal #A of a set A >>> smaller.by.one than sets fuller.by.one >>> #A ∈ ⟦0,ℵ₀⦆ :⇔ (#A < #(A∪{a}) ⇐ A ≠ A∪{a}) >>> >>> If, >>> as might be expected, >>> infinity.plus.one is different from simple.infinity, >>> then, >>> under that definition, >>> infinity is finite. >> >> It's "well-ordering the universe". > > Please complete this sentence: > ⎛ In "It's 'well-ordering the universe'", > ⎜ "it" refers to > ⎝ > >> Yeah, I know, >> you don't have a universe in your theory, >> as you say that >> there's no meta-theory your theory, >> yet, what's that then, all one theory? > > I think that a universeᴿꟳ and a universeⁿᵒᵗᐧᴿꟳ > are different. > > ⎛ In the formal sciences, the domain of discourse, > ⎜ also called the universe of discourse, universal set, > ⎜ or simply universe, > ⎜ is the set of entities over which > ⎜ certain variables of interest in some formal treatment > ⎝ may range. > https://en.wikipedia.org/wiki/Domain_of_discourse > > I have universesⁿᵒᵗᐧᴿꟳ for my theories, as I must > wherever there are variables, and > there are lots and lots of variables in 'my' theories. > > I take your universeᴿꟳ to be > a unique, all.inclusive universeⁿᵒᵗᐧᴿꟳ.and.domain. > > The logic (FOL) of variables and universesⁿᵒᵗᐧᴿꟳ > does not require an all.inclusive universeᴿꟳ > We only need to be able to talk about > what we are talking about, the current universeⁿᵒᵗᐧᴿꟳ, > whichever that is. > > > There are pragmatic motivations for talking about > an all.inclusive universeᴿꟳ. > > There are also pragmatic motivations for talking about > only what we are talking about, the current universesⁿᵒᵗᐧᴿꟳ. > > For example, if someone denies the existence of infinities, > a good place to start might be the universeⁿᵒᵗᐧᴿꟳ of finites, > which is itself not finite, and > which can disobey rules designed for finites. > >> there's no meta-theory your theory, >> yet, what's that then, all one theory? > > In these discussions, my bottom.floor logic is typically FOL, > the logic of variables and their universesⁿᵒᵗᐧᴿꟳ. > > My meta.theory of FOL is the theory of > finite sequences of claims, each claim of which is > true.or.not.first.false. > I think that I've mentioned that. > > So, is it, "not.ultimately.untrue"? See, here there's a meta-theory of all that, yet it's all in the theory, a heno-theory, otherwise being both formalist and platonic all the time. "The Logic", ? Is it, "not.ultimately.untrue"? You don't say, ....