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, 6 Feb 2025 09:59:02 +0100
Organization: A noiseless patient Spider
Lines: 25
Message-ID: <vo1tkm$2sbfs$1@dont-email.me>
References: <vmo1bs$1rnl$1@dont-email.me>
 <87e2e677c7802c9c17df6063f340cb5857d5700b@i2pn2.org>
 <vnd4h8$2c0st$1@dont-email.me>
 <c50fde56e7e0c4cf4842d4944ea3d1917c75eb41@i2pn2.org>
 <vnfftp$2rv3t$1@dont-email.me>
 <680d4249c9bf1504231a53732ac5096184261495@i2pn2.org>
 <vngumj$34ss1$6@dont-email.me> <12a38458-bfb9-4611-9072-eadbb166c0ec@att.net>
 <vnl5ll$3ae6$3@dont-email.me> <d47ddb72-2ab1-4923-b8db-2d01777f20ab@att.net>
 <vnnknt$knr7$7@dont-email.me> <e187d378-3c4a-4cf6-b57e-b8f623cac0e7@att.net>
 <vnqdi5$188h1$1@dont-email.me> <a007714a-1b8c-4d9a-9b90-64c37e2bdef3@att.net>
 <vnr2au$1cbur$2@dont-email.me> <908c8431-3d44-496c-8f5c-e33cc9554956@att.net>
 <vnsp4h$1phsq$1@dont-email.me> <faae7d60-6287-4734-87d8-091c3c3e0aa3@att.net>
 <vnva5b$2al9b$1@dont-email.me> <1ab7ff67-f1fb-4814-9d28-c883a4756097@att.net>
 <vo07bf$2fh7t$1@dont-email.me> <451804be-c49f-43ab-bca9-8a4af406d945@att.net>
 <vo0di8$2fh7t$2@dont-email.me>
 <b46a22eb014d24357179f7dc1d5e15fc2da2a102@i2pn2.org>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Thu, 06 Feb 2025 09:59:03 +0100 (CET)
Injection-Info: dont-email.me; posting-host="38d752fee4f88a2da06b35d50f0a18b6";
	logging-data="3026428"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18U5o2oLQEfJ4SfVLESVH+mPDp5LV/4yuE="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:qZrht8PZ1pBJ+2qVNQdrSh7Rn1c=
Content-Language: en-US
In-Reply-To: <b46a22eb014d24357179f7dc1d5e15fc2da2a102@i2pn2.org>
Bytes: 2726

On 06.02.2025 03:29, Richard Damon wrote:
> On 2/5/25 2:18 PM, WM wrote:
>> On 05.02.2025 19:43, Jim Burns wrote:
>>
>>> It might just matter what a natural number, induction,
>>> a FISON, and a union are.
>>
>> The axiom of induction:∀P( P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)))
>>
>> P(1): U(F(n) \ F(1)) = ℕ.
>>
>> P(k): U(F(n) \ {F(1), F(2), ..., F(k)}) = ℕ
>> ==>
>> P(k+1): U(F(n) \ {F(1), F(2), ..., F(k+1)}) = ℕ.
>>
> And thus you can claim that no F(n) is "necessary".

My proof is this: IF U(F(n)) = ℕ, THEN U({ }) = { } = ℕ.
> 
> Doesn't mean you can't use a set of them to build the set of Natural 
> Numbers.

It means precisely that. The premise is wrong because { } = ℕ is wrong.

Regards, WM