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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Newsgroups | sci.math |
| Subject | Re: Ordinals |
| Date | 2024-03-01 19:41 -0500 |
| Organization | A noiseless patient Spider |
| Message-ID | <ff9ab96d-a0b8-435e-8d1f-bd784c705ca4@att.net> (permalink) |
| References | (14 earlier) <urtblr$qfb8$1@solani.org> <urtckj$qfmv$1@solani.org> <4097b7df-ccbf-401b-aec3-91a746c1a533@att.net> <urtipg$qql1$1@solani.org> <urtjqo$qr8k$1@solani.org> |
On 3/1/2024 5:11 PM, Mild Shock wrote:
> About digging into
> "transfinite.induction is well.order in drag"
> by Jim Burns. You probably mean
> transfinite.induction follows from well.order.
> What about the other direction?
Both directions.
well.order
∃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:(̅p[0,β) ⇒ ̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β)∧̅p(β)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
∀ᵒʳᵈβ:(̅p[0,β) ⇒ ̅p[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:̅p(γ)
transfinite.induction
> Now my question, is assume we have no foundation,
> but epsilon induction, what will happen. epsilon
> induction is usually not an axiom. But what
> if we stipulate it as an axiom?
>
> Considered as an axiomatic principle, it is
> called the axiom schema of set induction.
> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
> https://en.wikipedia.org/wiki/Epsilon-induction
That wiki.page assumes regular sets.
I'm not sure it does so explicitly,
but it defines ordinals as
transitive sets of transitive sets
which, without regularity, include x = {x}
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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
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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
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