Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]
| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Newsgroups | sci.math |
| Subject | Re: Ordinals |
| Date | 2024-02-27 14:25 -0500 |
| Organization | A noiseless patient Spider |
| Message-ID | <e001f1ca-b302-46e2-a0f8-bb77508df7e2@att.net> (permalink) |
| References | (6 earlier) <G9GyyDplthr8FEUv73Z4g75-jek@jntp> <a91d312a-5782-4408-99a4-01e23dba936f@att.net> <MhlSS9xIp1LJFzMC_jl-sw-RPbo@jntp> <4485f066-345f-406b-a5c2-4139910cf512@att.net> <3xtILSwZ23f1EUFhZbWPwG47aFA@jntp> |
On 2/23/2024 3:47 AM, WM wrote: > Le 22/02/2024 à 17:13, Jim Burns a écrit : >> ...] > > Having three corners is essential for triangles. > Being well-ordered is essential for ordinals. > What I meant is that > we cannot follow the well-order into the dark realm. > In particular Peano ceases. "Following the well.order" is a metaphor. I am not a Form.. I've never met you. Still, I'm confident that you aren't a Form, either. We aren't in the same Realm as the ordinals. Un.metaphorically, in our Realm, we have no opportunity to follow or to not.follow the well.order. What I gather that you (WM) are asserting by metaphor is a wrong assertion. > Having three corners is essential for triangles. > Being well-ordered is essential for ordinals. > What I meant is that > we cannot follow the well-order into the dark realm. > In particular Peano ceases. We know that Peano induction not.ceases in the Peano (final) ordinals. It is a mistake to expect more than Peano induction not.ceasing in the Peano (final) ordinals. We know that transfinite induction not.ceases in the transfinite ordinals. It is a mistake to expect more than transfinite induction not.ceasing in the transfinite ordinals. We know that being well.ordered not.ceases in the transfinite ordinals. Even beyond the first _inaccessible_ ordinal κ being well.ordered not.ceases. https://en.wikipedia.org/wiki/Inaccessible_cardinal | In set theory, an uncountable cardinal is inaccessible | if it cannot be obtained from smaller cardinals by | the usual operations of cardinal arithmetic. We know that because transfinite induction is well.order in drag. Thus, just like well.order, transfinite induction not.ceases in the ordinals. ---- Well.ordering not.ceases in the ordinals. If exists ordinal γ: p(γ) then exists first ordinal β: p(β) ∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β) ∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ) ∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ) ∀ᵒʳᵈβ:(̅p(β) ∨ ¬∀ᵒʳᵈα<β:̅p(α)) ⟹ ∀ᵒʳᵈγ:̅p(γ) ∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:̅p(α) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ) ∀ᵒʳᵈβ:(̅pᣔ[0,β) ⇒ ̅pᣔ[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:̅p(γ) where ̅pᣔ[0,β) ⟺ ∀ᵒʳᵈα<β:̅p(α) ̅pᣔ[0,β) ∧ ̅p(β) ⟺ ̅pᣔ[0,β⁺¹) If, for each ordinal β ̅p.before.β implies ̅p.before.β⁺¹ then, for each ordinal γ ̅p Transfinite induction not.ceases in the ordinals.
Back to sci.math | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread
Re: Ordinals "markus...@gmail.com" <markusklyver@gmail.com> - 2024-02-20 11:35 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-20 20:54 +0100
Re: Ordinals "mitchr...@gmail.com" <mitchrae3323@gmail.com> - 2024-02-20 11:59 -0800
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 12:15 -0800
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 12:27 -0800
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-20 17:02 -0500
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 19:36 -0800
Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 05:47 -0800
Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 05:50 -0800
Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 06:16 -0800
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-21 10:24 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 09:03 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 09:11 +0100
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 10:16 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:20 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:22 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:40 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:00 +0100
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 10:55 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:08 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:13 +0100
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 11:01 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:11 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:13 +0100
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-22 14:34 -0500
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 11:39 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-23 00:41 +0100
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-21 08:33 +0000
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-21 12:59 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-22 13:00 +0000
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-22 11:13 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-23 08:47 +0000
Semanticists candy (Was: Ordinals) Mild Shock <janburse@fastmail.fm> - 2024-02-26 12:58 +0100
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-27 14:25 -0500
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-27 11:59 -0800
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-27 20:05 +0000
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-27 17:24 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-28 09:48 +0000
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-28 06:52 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-28 17:24 +0000
Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-28 17:07 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-29 08:23 +0000
Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-29 07:35 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-29 19:25 +0000
Re: Ordinals "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-02-29 13:48 -0800
Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-29 22:12 -0500
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-29 19:36 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 20:38 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 20:52 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 21:08 +0100
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-03-01 15:56 -0500
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 22:53 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 23:11 +0100
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 23:33 +0100
Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-03-01 19:41 -0500
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-01 20:15 -0800
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-02 10:49 -0800
Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-02 20:40 +0100
Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-01 14:45 -0800
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-01 08:47 +0000
Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-01 09:44 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-01 18:29 +0000
Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-01 14:00 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-02 12:36 +0000
Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-02 09:22 -0500
Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-02 15:25 +0000
Re: Ordinals "mitchr...@gmail.com" <mitchrae3323@gmail.com> - 2024-02-20 17:21 -0800
RSemanticists candy (Re: Ordinals) [Addendum] Mild Shock <janburse@fastmail.fm> - 2024-02-26 13:01 +0100
csiph-web