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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Sun, 16 Mar 2025 20:41:43 -0000 (UTC)
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Am Sun, 16 Mar 2025 17:17:33 +0100 schrieb WM:
> On 16.03.2025 13:17, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>> On 15.03.2025 12:57, Alan Mackenzie wrote:
>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>> 
>>>> I'm showing you that your "definition" of "definable numbers" is no
>>>> definition at all.
>>> You are mistaken. Not all numbers have FISONs because ∀n ∈ U(F): |ℕ \
>>> {1, 2, 3, ..., n}| = ℵo.
>>> ℵo numbers have no FISONs.
>> You haven't said what you mean by F.
> I did in the discussion with JB: F is the set of FISONs.
> 
>> All natural numbers "have" a FISON
> Then all natural numbers would be in FISONs. But because of ∀n ∈ U(F):
> |ℕ \ {1, 2, 3, ..., n}| = ℵo all FISONs fail to contain all natural
> numbers.
Not at all. N, equivalent to omega, is not a FISON.

>> If you really think there is a non-empty set of natural numbers which
>> don't "have" FISONs,
> Of course there is such a set. It contains almost all natural numbers.
No, there are already infinitely many countable ones (=with FISONs).

>> then please say what the least natural number in that set is, or at the
>> very least, how you'd go about finding it.
> The definable numbers are  potentially infinite sequence. With n also
> n+1 and n^n^n belong to it.
Yes, and omega is the least number larger than all those.

>>> The subtraction of all numbers which cannot empty ℕ cannot empty ℕ.
>> You've never said what you mean by a number "emptying" a set.
> Removing all its elements by subtraction.

>> It's unclear whether you mean the subtraction of each number
>> individually, or of all numbers together.
> If all natural numbers were individually definable, then there would not
> be a difference.
There is a big difference between subtracting one number from a set and
subtracting all numbers.

>> Even "subtraction" is a non-standard word, here.  The opposite of "add"
>> (hinzufügen) is "remove", not "subtract".
> The opposite of addition is subtraction. Look for instance:
> subtraction+of+sets+latex
imma subtract my latex set iykwim

>>>>>> It all depends on the X from which N_def is formed.  If X is N \
>>>>>> {1},
>>>>> Then its elements are mostly undefined as individuals.
>>>> "Undefined as individuals" is an undefined notion,
>>> No. It says simply that no FISON ending with n can be defined.
>> A FISON is a set.  Sets don't "end" with anything.
> A FISON is a well-ordered set or segment or sequence. It has a largest
> element.
>
>>>>> Every element has a finite FISON. ℕ is infinite.  Therefore it
>>>>> cannot be emptied by the elements of ℕ_def and also not by ℕ_def.
>>>> A "finite" FISON?  What other type is there?
> None, but you should pay attention because ℕ is infinite and therefore
> cannot be emptied by finite sets.
It can be emptied by the infinitely many sets {k} for every k in N.

>>>> What do you mean by "having" a FISON?  What does it mean to "empty" N
>>>> by a set or elements of a set?  What is the significance, if any, of
>>>> being able to "empty" a set?
>>> Simply try to understand. I have often stated the difference:
>>> ∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo ℕ \ {1, 2, 3, ...} = { }
>> Which doesn't address my question in the sightest.  What do you mean by
>> "emptying" N by a set or by elements of a set?
> Subtracting a set or elements of a set. See above. Definable elements
> can be subtracted individually. Undefinable elements can only be
> subtracted collectively.
>
>> You haven't said what (if anything) you mean by a number emptying N. 
>> And every natural number "has" a FISON, not just some subset of them.
> You seem unable to learn.
>> 
>>> They are placed on the ordinal line and can tend to ℕ. This can happen
>>> only on the ordinal line.  Your assertion of the contrary is therefore
>>> wrong.
>> Of the many assertions I've made, the one you're referring to is
>> unclear.
> You said: The tending takes place, but not in a "place".
>> 
>>>> "Defined numbers" appears not to be a coherent mathematical concept. 
>>> The subtraction of all numbers which cannot empty ℕ cannot empty ℕ.
>>> The collection of these numbers is ℕ_def.
>> Incoherent garbage.
> You really have problems to comprehend sentences. Try again.
No, you try reformulating.

>> You haven't said what you mean by a number "emptying" a set.
> Even if I had not, an intelligent reader would know it.
It is your responsibility to make yourself understood.

>> The current state of our discussion is that you have failed to give any
>> coherent definition of "defined numbers";
> A defined number is a number that you can name such that I understand
> what you mean. In every case you choose almost all numbers will be
> greater.
And they will all be "defined".

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.