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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sun, 02 Mar 2025 14:28:28 -0500
Organization: Peripheral Visions
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WM expressed precisely :
> On 02.03.2025 14:41, FromTheRafters wrote:
>> WM brought next idea :
>>> On 01.03.2025 19:38, FromTheRafters wrote:
>>>> WM presented the following explanation :
>>>
>>>>> Processes using defined individuals like FISONs cannot surpass a finite 
>>>>> set. FISONs are finite. Their number will never be greater than finite.
>>>>> {1}
>>>>> {2, 1}
>>>>> {3, 2, 1}
>>>>> ...
>>>>
>>>> Those are still not FISONs.
>>>
>>> They are. The order in curly brackets does not matter.
>> 
>> AI Chatbot
>> 
>> https://poe.com/chat/370x9xpc7krhbconr8j
>> 
>> Assistant
>> Poe
>> 
>> The term "FISON" stands for "finite initial segment of natural numbers." A 
>> FISON is typically defined as a finite set of the form {0,1,2,…,n} for some 
>> non-negative integer n.
>> 
>> The set {4,3,2,1} does not meet this definition for a couple of reasons:
>> 
>>     Order: A FISON must include all natural numbers starting from 0 up to n 
>> in ascending order. The set {4,3,2,1} is not in ascending order and does 
>> not include 0.
>> 
>>     Completeness: A FISON must include all natural numbers less than or 
>> equal to its maximum element. In this case, the maximum element is 4, but 
>> the set is missing 0.
>> 
>> Therefore, {4,3,2,1} is not a FISON. A valid FISON that includes up to 4 
>> would be {0,1,2,3,4}.
>
> Two mistakes. FISONs have been created by Virgil and me. Therefore we know 
> what their definition is. They start with 1. Ascending order is not a feature 
> of sets but of sequences.

Yes, but your notation of {3,2,1} is not a sequence but a set. A 
sequence would be (3,2,1). You write sets and try to treat them as 
segments of sequences.