Deutsch   English   Français   Italiano  
<vppfol$3280b$1@dont-email.me>

View for Bookmarking (what is this?)
Look up another Usenet article

Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Thu, 27 Feb 2025 11:45:41 +0100
Organization: A noiseless patient Spider
Lines: 94
Message-ID: <vppfol$3280b$1@dont-email.me>
References: <vmo1bs$1rnl$1@dont-email.me> <vp6rg8$2o8sd$7@dont-email.me>
 <9a88665f-211f-4260-b585-97c72c7b6d1b@att.net> <vp7qea$2tofq$5@dont-email.me>
 <8bed122d8b355eff96158e6f5cb76cffcc42925c@i2pn2.org>
 <vp9hl1$3afuk$6@dont-email.me>
 <7a26856916099747e76314a2b4c79693e14426fd@i2pn2.org>
 <vpacfc$3fush$1@dont-email.me> <b3d7070e-1381-41e3-9ace-0f21bc052d0b@att.net>
 <vpc885$3t9g8$1@dont-email.me> <e19b21b4-07f1-46ac-94f0-dac6cd114754@att.net>
 <vpetjf$eusv$3@dont-email.me> <98baf83e-820e-4e1b-be2c-e5ea4802683d@att.net>
 <vpfc74$hq5c$2@dont-email.me> <0876c2b9-2144-44c1-a26b-20176f5e2127@att.net>
 <vpft4t$kdg0$4@dont-email.me> <067f772a-4f4c-4c27-8042-3f605f814876@att.net>
 <vpi1g6$14ivq$7@dont-email.me> <vpi7eu$17stc$1@dont-email.me>
 <vpi9jo$18qai$2@dont-email.me> <fa7bb863-570e-4602-b932-277b01133bba@att.net>
 <vpk0nn$1s04m$1@dont-email.me> <dd62224a-579b-4032-be2c-04c305247753@att.net>
 <vpmvg3$2i1ev$1@dont-email.me> <558a879a-4130-476a-8b5d-d53cd371919b@att.net>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Thu, 27 Feb 2025 11:45:42 +0100 (CET)
Injection-Info: dont-email.me; posting-host="c186eda793122898af663c392b76d5f9";
	logging-data="3219467"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1/HoglixdKER0XCmWdXpFdv4lm93Ei1+gQ="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:poO+gRQPd+/VWpHf4sETXGGmXG8=
Content-Language: en-US
In-Reply-To: <558a879a-4130-476a-8b5d-d53cd371919b@att.net>
Bytes: 5304

On 26.02.2025 23:17, Jim Burns wrote:
> On 2/26/2025 6:55 AM, WM wrote:
>> On 25.02.2025 19:39, Jim Burns wrote:
> 
>>>
>>
>> Zermelo for instance made such claims:
>> Um aber die Existenz "unendlicher" Mengen zu sichern,
>> bedürfen wir noch des folgenden, seinem wesentlichen Inhalte
>> von Herrn Dedekind herrührenden Axioms.
>> ... Der Bereich enthält mindestens eine Menge Z, welche die Nullmenge 
>> als Element enthält und
>> so beschaffen ist, daß jedem ihrer Elemente a
>> ein weiteres Element der Form {a} entspricht
>> ... Die Menge Z_0 enthält die Elemente 0, {0}, {{0}}, usw.
>> und möge als "Zahlenreihe" bezeichnet werden,
>> ... Sie bildet das einfachste Beispiel einer
>> "abzählbar unendlichen" Menge.
>> [E. Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I,
>> Mathematische Annalen (1908), S. 266]
> 
> ⎛ But in order to ensure the existence of "infinite" sets,
> ⎜ we still need the following axiom,
> ⎜ the essential content of which comes from Mr. Dedekind.
> ⎜ ... The domain contains at least one set Z,
> ⎜ which contains the zero set as an element and
> ⎜ is such that each of its elements a
> ⎜ corresponds to another element of the form {a}
> ⎜ ... The set Z_0 contains the elements 0, {0}, {{0}}, etc.
> ⎜ and may be referred to as a "number series", ...
> ⎜ It forms the simplest example of a "countably infinite" set.
> ⎜ [E. Zermelo: Investigations on the Foundations of Set Theory I,
> ⎜ Mathematische Annalen (1908), p. 266]
> ⎝
> -- google

Good. Much better than 10 years ago.
> 
> The domain contains at least one set Z.
> 0∈Z ∧ ∀a:a∈Z⇒{a}∈Z
> 
> It is indefinite which set Z refers to,
> apart from that claim definitely being true of Z.

The set containing 1 and with every x also x+1 may be ℚ, ℝ, or ℂ. 
Important is that it contains ℕ.
> 
> That set Z has a subset Z₀ which might not be Z
> (what is Z, after all?)
> which may be referred to as a number series.
> 
> The number series Z₀ holds no extra numbers.
> Suppose set W and Y are candidates for Z
> 0∈W ∧ ∀a:a∈W⇒{a}∈W
> 0∈Y ∧ ∀a:a∈Y⇒{a}∈Y
> and
> some elements are in W but not in Y.
> Those elements in W\Y are extra, and not.in Z₀

The specific form of Z is irrelevant.
Z₀ is the Zahlenreihe, the set {0, 1, 2, ...}
> 
> Perhaps, outside the material you quote,
> Zermelo has said that
> Z₀ holds no extra numbers
> or some version of that.
> I haven't investigated what Zermelo has said.
> Whatever he's said, the claims I've made here are true.

Yes.

> Because Z₀ holds no extra numbers,
> Z₀ is definite, even though superset Z is indefinite.
> That is, in my opinion, what makes all this work.

Z₀ is ensured by induction. That is the point.
> 
> This next bit you (WM) might like, for a change.
> It looks like the pseudo.induction.rule which
> you have been trying to use.

It is induction.
> Z₀ being the unique intersection of inductive subsets
> makes
> Z₀ its.own.only.inductive.subset,
> which makes
> Z₀ the only correct set to use in a proof by induction.

It is proved to exist by induction, i.e., by the rule that 0 exists and 
with a also a'.

By the same induction I prove UF = ℕ  ==> Ø = ℕ.

Regards, WM