Path: ...!local-2.nntp.ord.giganews.com!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Sun, 11 Aug 2024 23:40:42 +0000 Subject: Re: Replacement of Cardinality (infinite middle) Newsgroups: sci.logic,sci.math References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@jntp> <IzWzFdkkm97GEXyAioF3IpRiSfI@jntp> <42d2b329-5394-47e0-b8c9-098908b2e9a8@att.net> <__cCn6h6Ey1Kz0BrIf6EShypg4M@jntp> <e8a3a66a-7d83-4658-9f4c-23d7dc354fb9@att.net> <iqelfxYKWhBbwcm10DcO5hr3scI@jntp> <f920592b-897c-48b9-a9af-80f25bc60e4b@att.net> <DDPks1ynTy6IhIWNHaxt25GM1v0@jntp> <c1f0efc8-04ca-4f2d-9820-cfd54c0eca73@att.net> <v90rp5$3dbpd$1@dont-email.me> <L8Pl0ELcnLfKVO0KrMmhSqDd-Y0@jntp> <v926ot$3tjq6$1@dont-email.me> <PbydnQO1H-qe_yj7nZ2dnZfqnPqdnZ2d@giganews.com> <v98ppi$ue8i$1@dont-email.me> <v98qeo$ul4c$1@dont-email.me> <v98qm3$ue8i$4@dont-email.me> <v98rnd$uuqh$1@dont-email.me> <v9926l$102t7$1@dont-email.me> <v993ip$10cor$1@dont-email.me> <v99e6j$1etn8$1@dont-email.me> <v9anp7$2o5mp$2@dont-email.me> <ad6dnanB9ZgbYyX7nZ2dnZfqnPSdnZ2d@giganews.com> <5f795e1a-346b-43f7-a2d2-7844591f5296@att.net> From: Ross Finlayson <ross.a.finlayson@gmail.com> Date: Sun, 11 Aug 2024 16:39:52 -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: <5f795e1a-346b-43f7-a2d2-7844591f5296@att.net> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: <-oGdnWXm-ZVn1iT7nZ2dnZfqn_ednZ2d@giganews.com> Lines: 130 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-Mv9Yca6Kd80DkJhQOR73pLw0n5NGfl3VVPUAz3yXZywTNYNs650bwYR/Dgd87+g64RMi6aL70xWy/8w!7qzj/iyo1eEd1zkuVz8f1v0jtwxW8aRjTUu6GXNn/41OYECUXzabGQVcwcxBe0Hyppus+QrH0QJG!7g== 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: 6009 On 08/11/2024 02:38 PM, Jim Burns wrote: > On 8/11/2024 2:10 PM, Ross Finlayson wrote: >> On 08/11/2024 09:10 AM, Moebius wrote: > >>> [...] >> >> How do you see omega >> as the second constant after empty set >> an inductive set in ZF? >> It's definitely not "all" infinity. > > ω is defined to be (the set of) all finite ordinals. > In that sense, ω is all of finiteness. > > ω is followed by all the transfinite ordinals. > In that sense, ω is not all of infinity. > Nearly none of it, really. > > ---- > U and V are inductive sets. > > ⋂{ind:U} is the intersection of inductive U.subsets. > ⋂{ind:U} is inductive. > for each inductive A ⊆ U: ⋂{ind:U} ⊆ A ⊆ U > > ⋂{ind:V} is the intersection of inductive V.subsets. > ⋂{ind:V} is inductive. > for each inductive B ⊆ V: ⋂{ind:V} ⊆ B ⊆ V > > In particular, U∩V is an inductive U.subset and V.subset. > > As an inductive V.subset, > ⋂{ind:V} ⊆ U∩V ⊆ V > > As an inductive U.subset, > ⋂{ind:U} ⊆ ⋂{ind:V} ⊆ U∩V ⊆ U > > ⋂{ind:U} ⊆ ⋂{ind:V} > Similarly, > ⋂{ind:V} ⊆ ⋂{ind:U} > ⋂{ind:U} = ⋂{ind:V} > > ⋂{ind:U} = ⋂{ind:V} := ⋂{ind} > the unique intersection of inductive subsets. > > ---- > {fin} is the set of finite ordinals. > ⋂{ind} is the intersection of inductive subsets. > > There is no first finite.ordinal ∉ ⋂{ind} > There is no finite ordinal ∉ ⋂{ind} > {fin} ⊆ ⋂{ind} > > Each inductive set ⊇ ⋂{ind} > {fin} is inductive. > {fin} ⊇ ⋂{ind} > > {fin} ⊆ ⋂{ind} > {fin} ⊇ ⋂{ind} > {fin} = ⋂{ind} := ω > >> How do you see omega > > ω is the set of all finite ordinals and is, > for each inductive set, > the intersection of inductive subsets. > > https://www.youtube.com/watch?v=6liqzvagSb8 Here's some good listening, a philosophy professor explains why metaphysics is logically necessary and may be coherent, and that the rejection thereof is irrational, or as Spock might put it, "illogical". He makes good points about the "indivisibility of truth" and that a conscientious theory has a metaphysical association with truth "de re" and "de res" not just "de dicto". Starting with a theory _without_ the constant introduced named omega, i.e., finite sets, given that there's axiomatized well-foundedness when otherwise simple comprehension would make the "omega" into an extra-ordinary or non-well-founded or inconsistent-multiplicity of a set, starting _without_ omega, the finite sets like ordinals, are, exactly those sets that don't contain themselves. Then, omega, as you've defined it, contains itself, again just quantifying over the specification of what omega purports to be, including your hereditarily finite build-up which is the same thing. So, even with your Russellian retro-thesis, even with "mere induction", even with axioms otherwise, you're stuck again with both Russell and Frege, regardless which you pick. So, omega the constant of ZF or ZFC is just a fragment or extension, it's a fixed-point, and yes this is correct usage, in a deconstructive account of your received theory there, why induction doesn't get a say. So, "omega is not all of infinity", whether it's +-1, besides the entire cumulative hierarchy, of transfinite cardinals. There are models of integers where there aren't standard models of integers. I'm curious, now that you have a beginning and an end of the finite, or 0 and omega in ZF, first of all isn't that a fixed-point and compactification also, as of about the same thing as a point-at-infinity for either perspective in geometry or fixed-point in number theory, one of the regular singular points of the hypergeometric the 0, 1, infinity? Second, do you just say "see Russell's rule"? Third, didn't you already have one? Isn't there already one? If not, where did your otherwise sound naive induction end?