Path: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Sun, 09 Mar 2025 17:41:02 +0000 Subject: Re: The set of necessary FISONs Newsgroups: sci.math References: <8e7b322e-1259-4563-b2d5-37983249a397@att.net> <4ae7b6d4-49a9-47ba-b2ac-c77238e93545@att.net> <60614d74-cf15-4e8e-8390-f8861bff44f9@att.net> <76d99693-1dcf-4049-98b9-a33edced2e83@att.net> <123fb080-4f72-482c-a6e9-aa525aa7150b@att.net> <00fb52fc-ca18-4166-90c7-71b5a66e2dda@att.net> From: Ross Finlayson Date: Sun, 9 Mar 2025 10:41:05 -0700 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: <00fb52fc-ca18-4166-90c7-71b5a66e2dda@att.net> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 119 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-uBHQTOlFA0V4xeww2EgDd1s7LOVnJbkfD5GCErLw6937mgiqMd/8Yje2DQrWrJOV9xbSdb14ZFs9Oat!+FY2NOD2fw3JTzomUiz7N6sRwYoMZnjtL1oIi7Yv1S1Wpl/bVLr0ZnDxjEKbL4tXP4NKcpgRSZ8= 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: 5478 On 03/09/2025 09:26 AM, Jim Burns wrote: > On 3/9/2025 5:55 AM, WM wrote: >> On 08.03.2025 20:27, Jim Burns wrote: >>> On 3/8/2025 9:09 AM, WM wrote: >>>> On 08.03.2025 12:58, Jim Burns wrote: > >>>>> you should have said somewhere >>>>> what a finite ordinal is, >>>> >>>> That is not under discussion here. >>> >>> That has been under discussion for decades. >>> >>> I think that these decades of discussion >>> have been, in large part, you assigning >>> different meanings to 'finite', etc. >>> and matheologians (among whom I place myself) >>> trying to discern what your meanings are. >>> >>> Here's my best guess: >>> definableᵂᴹ == finite > > Compare what you quote to what I wrote: > definableᵂᴹ == finiteⁿᵒᵗᐧᵂᴹ == #A<#Aᣕᵇ > >> [...] >> >>> darkᵂᴹ == finite == big > > Compare what you quote to what I wrote: > darkᵂᴹ == finiteⁿᵒᵗᐧᵂᴹ == big and #A<#Aᣕᵇ > matheologicalᵂᴹ == infiniteⁿᵒᵗᐧᵂᴹ == #A=#Aᣕᵇ > >> [...] >> >>>> n is usually denoting >>>> a natural number. >>> >>> Do we mean the same by 'natural number'? >> >> There are two different meanings: >> All positive integers having FISONs or all positive integers. > > The distinction which you (WM) > have been dodging for decades > is between > sets which have > fuller.by.one and emptier.by.one counterparts > _which are a different size_ (finiteⁿᵒᵗᐧᵂᴹ) > and > sets which have > fuller.by.one and emptier.by.one counterparts > _which are the same size_ (infiniteⁿᵒᵗᐧᵂᴹ) > >>>> FISONs are finite by definition. >>> >>> Do we mean the same by 'finite'? >> >> A natural number n is finite. >> It is an integer between 0 and ω: 0 < n < ω. >> Sometimes 0 is included, never ω is included. > > You erased the distinction I made and > you haven't given any replacement. > (Your ωᵂᴹ is merely a big finite.) > > #A size of set A > #Aᣕᵇ size of fuller.by.one set, Aᣕᵇ = A∪{b} ≠ A > > #A < #Aᣕᵇ larger fuller.by.one set: A finite > > #Y = #Yᣕᶻ same.size fuller.by.one set: Y infinite > > The distinction infinite.v.finite is useful > because > not all sets are finite, > and the distinction, for fuller.by.one sets, > of same.v.different sizes leads to > very different properties. > > A notable example of an infinite (ie, #Y = #Yᣕᶻ) set > is the set of finite set.sizes {#A:#A<#Aᣕᵇ} > > ⎛ For each finite.set.size ξ in {#A:#A<#Aᣕᵇ}, > ⎜ there is a larger subset #{#A<ξ+1:#A<#Aᣕᵇ} = ξ+1 > ⎜ > ⎜ {#A:#A<#Aᣕᵇ} doesn't contain > ⎜ a subset larger than {#A:#A<#Aᣕᵇ} > ⎜ > ⎜ For each finite.set.size ξ in {#A:#A<#Aᣕᵇ}, > ⎝ ξ isn't the size of {#A:#A<#Aᣕᵇ} > > {#A:#A<#Aᣕᵇ} is what we mean by ℕ, > what you mean by ℕ_def. > > ---- >>>> My description is definite. >>> >>> Your description is only definite because >>> ℕ ∋ n is definite. >> >> No, my description is definite >> because >> every n can be obtained by addition of 1's >> (or of curly brackets). > > Finitely.many 1's or curly.brackets. > How many is that? > > Your decades.long argument has been about > how many that is -- > but you avoid saying what you mean. > You even avoid hearing what we mean. > > Seems kind of futile doesn't it.