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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Newsgroups | sci.math |
| Subject | Re: how |
| Date | 2024-06-18 13:12 -0400 |
| Organization | A noiseless patient Spider |
| Message-ID | <c178dd0f-4bb0-47d9-b1e2-e8a7c8b851c0@att.net> (permalink) |
| References | (6 earlier) <5fWjlqMjzUHFygAk1yVCvtuDLOM@jntp> <067c2a88-553e-4eb5-9ead-efb0e9a39d43@att.net> <VIhO2Ae_yrPsVjSsDz1yJZy_E4g@jntp> <7792d74c-4ae9-4909-81cc-7d9975e8d510@att.net> <xJmCht9ieNLiQMSR57t03IZLuXs@jntp> |
On 6/16/2024 10:17 AM, WM wrote:
> Le 14/06/2024 à 20:52, Jim Burns a écrit :
>> every number in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
>> has ℵ₀ successors in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
>>
>>> Note that
>>> only the numbers with successors are deleted,
>>> the successors remain by definition.
>>
>> sⱼ∘f:{0<} → {j⁺¹<} is 1.to.1
minimal inductive meta.set {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
0.followers {0<} = {i:∀₃Xᴬ⤾⁺¹₀:X∋i ∧ 0<i}
j⁺¹.followers {j⁺¹<} = {i:∀₃Xᴬ⤾⁺¹₀:X∋i ∧ j⁺¹<i}
Non.empty subset S ⊆ {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
holds a first element k₁ with
predecessor j₁ = k₁⁻¹ unless k₁ = 0
j₁ ∉ S ∧ j₁⁺¹ ∈ S
Define
∃f:{0<}⇉{j<} ⇔
exists f:{0<} → {j<}: 1.to.1
{j<} holds at least as many as {0<}
|{0<}| ≤ |{j<}|
Define
S = {i:∀₃Xᴬ⤾⁺¹₀:X∋i ∧ ¬∃f:{0<}⇉{i<}}
S is the set of numbers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
without ℵ₀.many followers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
S is empty or nonempty.
If S is empty
then
NO number in {i:∀₃Xᴬ⤾⁺¹₀:X∋i} is
WITHOUT ℵ₀.many followers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
Deleting all numbers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
with ℵ₀.many followers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
deletes all numbers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
If S is nonempty
then
S holds a first element k₁ with
predecessor j₁ = k₁⁻¹ unless k₁ = 0
j₁ ∉ S ∧ j₁⁺¹ ∈ S
∃f:{0<}⇉{j₁<} ∧ ¬∃g:{0<}⇉{j₁⁺¹<}
However,
sⱼ₁:{j₁<}⇉{j₁⁺¹<}:1.to.1
sⱼ₁∘f:{0<}⇉{j₁⁺¹<}:1.to.1
∃g=sⱼ₁∘f:{0<}⇉{j₁⁺¹<}:1.to.1
Contradiction.
Therefore,
S the set of numbers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
without ℵ₀.many followers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
is empty
and
deleting all numbers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
with ℵ₀.many followers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
deletes all numbers in {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
> If only numbers having ℵo successors are removed
> and only as long as ℵo successors remain,
> then ℵo successors remain and
> every definable number is removed.
> You cannot defina a number that remains.
> But ℵo successors remain
If all numbers having ℵ₀ followers are removed
then no numbers remain.
There is no first number remaining.
There is no number remaining.
We can _define_ a number that remains; however,
the _existence_ of the number defined
leads to self.contradiction
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Re: how Jim Burns <james.g.burns@att.net> - 2024-06-21 17:01 -0400
Re: how (running constants in physics) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-22 15:26 -0700
Re: how (running constants in physics) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-24 17:56 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-21 06:59 +0000
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-21 13:32 -0400
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-21 15:13 -0400
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-22 07:58 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 16:31 +0000
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-22 14:20 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 20:19 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-22 13:22 -0700
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-22 18:17 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 12:11 +0000
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-22 13:34 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 20:12 +0000
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-22 17:04 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 21:30 +0000
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-23 13:58 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-23 20:04 +0000
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-24 13:18 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-24 18:41 +0000
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-22 18:21 -0400
Re: how Phil Carmody <pc+usenet@asdf.org> - 2024-06-23 01:25 +0300
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-23 19:52 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-22 12:09 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 20:23 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-22 14:16 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-22 14:18 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 21:35 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-22 14:41 -0700
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-22 17:25 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-22 21:34 +0000
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 13:45 +0000
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-13 08:22 -0400
Re: how Moebius <invalid@example.invalid> - 2024-06-13 14:50 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 14:39 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 16:53 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 15:07 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 17:23 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 14:28 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 16:40 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 15:01 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 17:09 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-13 17:35 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 15:38 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 15:39 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-10 13:44 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-10 15:58 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 15:33 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 15:37 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-11 00:47 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 15:49 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-11 01:09 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 16:15 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-11 03:28 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-11 03:41 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-11 03:43 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-11 12:31 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-12 01:33 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 16:20 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-11 01:28 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 16:54 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-11 03:07 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-11 12:41 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-11 12:43 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-11 13:00 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-12 01:24 +0200
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-12 06:50 -0400
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-12 06:49 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-10 13:33 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-10 16:03 +0200
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-10 13:43 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-10 17:50 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-10 22:14 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 15:40 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-11 00:50 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-06 10:52 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-07 10:59 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-07 11:59 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-07 12:00 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-12 13:18 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-12 20:38 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-12 13:51 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-12 13:57 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-12 21:01 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 04:37 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 14:04 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 16:32 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 14:56 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 17:00 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 15:10 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-13 11:57 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-13 11:56 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-14 08:41 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-14 11:30 -0700
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-14 17:21 -0400
Re: how Moebius <invalid@example.invalid> - 2024-06-14 23:42 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-14 23:43 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-16 14:21 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-16 12:05 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-16 12:11 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-16 21:41 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-16 21:51 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-17 00:52 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-16 17:11 -0700
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-17 20:04 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-18 20:13 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-18 15:21 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-19 00:45 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-18 16:18 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-19 01:47 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-18 18:05 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-19 07:42 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-19 12:36 -0700
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-18 20:45 -0400
Re: how Moebius <invalid@example.invalid> - 2024-06-19 02:56 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-18 18:02 -0700
Re: how (quantities and units, implicits and explicits, intensional and extensional) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-18 19:34 -0700
Re: how (quantities and units, implicits and explicits, intensional and extensional) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-18 20:58 -0700
Re: how (quantities and units, implicits and explicits, intensional and extensional) Jim Burns <james.g.burns@att.net> - 2024-06-19 12:43 -0400
Re: how (quantities and units, implicits and explicits, intensional and extensional) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-19 13:29 -0700
Re: how (quantities and units, implicits and explicits, intensional and extensional) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-19 17:32 -0700
Re: how (quantities and units, implicits and explicits, intensional and extensional) Jim Burns <james.g.burns@att.net> - 2024-06-20 12:47 -0400
Re: how (quantities and units, implicits and explicits, intensional and extensional) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-20 13:40 -0700
Re: how (quantities and units, implicits and explicits, intensional and extensional) Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-06-20 18:27 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-16 17:13 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-17 21:19 +0000
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-12 20:58 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-12 14:03 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-12 14:04 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 10:48 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-13 14:28 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-13 14:31 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-13 11:53 -0700
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-14 08:40 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-14 11:30 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-14 22:43 +0200
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-12 20:16 -0400
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-13 11:59 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-13 01:47 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-06 20:11 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-06 20:12 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-07 11:01 +0000
Re: how Moebius <invalid@example.invalid> - 2024-06-07 13:06 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-07 13:10 +0200
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-07 15:57 +0000
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-07 13:32 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-07 19:15 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-07 12:27 -0700
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-07 17:49 -0400
Re: how Moebius <invalid@example.invalid> - 2024-06-07 23:59 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-07 22:45 -0700
Re: how FromTheRafters <FTR@nomail.afraid.org> - 2024-06-08 07:33 -0400
Re: how WM <wolfgang.mueckenheim@tha.de> - 2024-06-08 16:23 +0000
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-10 19:00 -0700
Re: how Moebius <invalid@example.invalid> - 2024-06-07 23:54 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-05 22:59 +0200
Re: how Jim Burns <james.g.burns@att.net> - 2024-06-05 17:46 -0400
Re: how Moebius <invalid@example.invalid> - 2024-06-06 01:08 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-06 01:11 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-06 01:13 +0200
Re: how Moebius <invalid@example.invalid> - 2024-06-06 01:18 +0200
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-03 13:04 -0700
Re: how "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-06-03 13:06 -0700
csiph-web