Choose a natural number as large as you can. Afterwards you can choose a
larger natural number. Why couldn't you choose this number before? What is
the reason that this holds for every choice?

Regards, WM

On 3/3/24 2:29 PM, WM wrote:
> Choose a natural number as large as you can. Afterwards you can choose a
> larger natural number. Why couldn't you choose this number before? What
> is the reason that this holds for every choice?
>
> Regards, WM

Why SHOULD we choose the highest possible number?

Since every Natural Number has a higher number, of course we can always
find a Higher number, that just comes out of the definition of the
Natural Numbers, EVERY Natural number has a successor, so there is aways
a choice for a higher number.

On 03.03.2024 21:57, Richard Damon wrote:
> On 3/3/24 2:29 PM, WM wrote:
>> Choose a natural number as large as you can. Afterwards you can choose
>> a larger natural number. Why couldn't you choose this number before?
>> What is the reason that this holds for every choice?
>>
>> Regards, WM
>
> Why SHOULD we choose the highest possible number?

It would be a good exercise to see that not all numbers are available.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> Since every Natural Number has a higher number, of course we can always
> find a Higher number, that just comes out of the definition of the
> Natural Numbers, EVERY Natural number has a successor, so there is aways
> a choice for a higher number.

But if the set ℕ is complete and no number is missing, why cannot every
number of ℕ be chosen at the first try?

Regards, WM

On 3/3/24 4:40 PM, WM wrote:
> On 03.03.2024 21:57, Richard Damon wrote:
>> On 3/3/24 2:29 PM, WM wrote:
>>> Choose a natural number as large as you can. Afterwards you can
>>> choose a larger natural number. Why couldn't you choose this number
>>> before? What is the reason that this holds for every choice?
>>>
>>> Regards, WM
>>
>> Why SHOULD we choose the highest possible number?
>
> It would be a good exercise to see that not all numbers are available.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

No, because if we TRY to pick the "highest" number, we find there is no
such number, because there is always a higher.

>>
>> Since every Natural Number has a higher number, of course we can
>> always find a Higher number, that just comes out of the definition of
>> the Natural Numbers, EVERY Natural number has a successor, so there is
>> aways a choice for a higher number.
>
> But if the set ℕ is complete and no number is missing, why cannot every
> number of ℕ be chosen at the first try?

We CAN choose every number, there just isn't a highest.

You just can't imagine how that works because your logic, and you brain,
are stuck in too small of a system.

That is a problem with YOU, not the numbers.

>
> Regards, WM
>

On 03.03.2024 22:53, Richard Damon wrote:
> On 3/3/24 4:40 PM, WM wrote:
>> On 03.03.2024 21:57, Richard Damon wrote:
>>> On 3/3/24 2:29 PM, WM wrote:

>>>> Choose a natural number as large as you can. Afterwards you can
>>>> choose a larger natural number. Why couldn't you choose this number
>>>> before? What is the reason that this holds for every choice?
+ >>> Why SHOULD we choose the highest possible number?
>>
>> It would be a good exercise to see that not all numbers are available.
>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> No, because if we TRY to pick the "highest" number, we find there is no
> such number, because there is always a higher.

We do not try to pick the highest number. The question is: Why can't we
pick immediately what we can pick later?

>> But if the set ℕ is complete and no number is missing, why cannot
>> every number of ℕ be chosen at the first try?
>
> We CAN choose every number, there just isn't a highest.

But we cannot choose in the first attempt what we can choose in the second
or third. What is the reason if not potential infinity, i.e., non
permanent existence of numbers?

Regards, WM

On 2024-03-03 19:29:42 +0000, WM said:

> Choose a natural number as large as you can. Afterwards you can choose
> a larger natural number. Why couldn't you choose this number before?
> What is the reason that this holds for every choice?

Lack of exercise.

--
Mikko

Le 04/03/2024 à 11:26, Mikko a écrit :
> On 2024-03-03 19:29:42 +0000, WM said:
>
>> Choose a natural number as large as you can. Afterwards you can choose
>> a larger natural number. Why couldn't you choose this number before?
>> What is the reason that this holds for every choice?
>
> Lack of exercise.

Somehow you are correct.
It is similar to the question why we are restricted in choosing large
prime numbers.

Regards, WM

On 3/4/24 3:58 AM, WM wrote:
> On 03.03.2024 22:53, Richard Damon wrote:
>> On 3/3/24 4:40 PM, WM wrote:
>>> On 03.03.2024 21:57, Richard Damon wrote:
>>>> On 3/3/24 2:29 PM, WM wrote:
>
>>>>> Choose a natural number as large as you can. Afterwards you can
>>>>> choose a larger natural number. Why couldn't you choose this number
>>>>> before? What is the reason that this holds for every choice?
> +
>>>> Why SHOULD we choose the highest possible number?
>>>
>>> It would be a good exercise to see that not all numbers are available.
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>>
>> No, because if we TRY to pick the "highest" number, we find there is
>> no such number, because there is always a higher.
>
> We do not try to pick the highest number. The question is: Why can't we
> pick immediately what we can pick later?

We could have if we wanted.

Who says you couldn't?

You can only blame yourself for not choising the number you wanted the
first time.

>
>>> But if the set ℕ is complete and no number is missing, why cannot
>>> every number of ℕ be chosen at the first try?
>>
>> We CAN choose every number, there just isn't a highest.
>
> But we cannot choose in the first attempt what we can choose in the
> second or third. What is the reason if not potential infinity, i.e., non
> permanent existence of numbers?

Again, why couldn't you choose that number first? What stopped you.

Your LATER sets of numbers have restrictions, that they can't be too
low, but nothing stopped you from choosing that higher number first.

You logic is just broken to say you couldn't

>
> Regards, WM

Le 05/03/2024 à 13:27, Richard Damon a écrit :
> On 3/4/24 3:58 AM, WM wrote:

>> Why can't we
>> pick immediately what we can pick later?
>
> We could have if we wanted.

Why didn't you want?
>
> Who says you couldn't?

Mathematics: ∀n ∈ ℕ_chosen: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>>
>>> We CAN choose every number,

No. ∀n ∈ ℕ_chosen: |ℕ \ {1, 2, 3, ..., n}| = ℵo

>> But we cannot choose in the first attempt what we can choose in the
>> second or third. What is the reason if not potential infinity, i.e., non
>> permanent existence of numbers?
>
> Again, why couldn't you choose that number first? What stopped you.

You are asked why you can't.
>
> Your LATER sets of numbers have restrictions, that they can't be too
> low, but nothing stopped you from choosing that higher number first.

∀n ∈ ℕ_chosen: |ℕ \ {1, 2, 3, ..., n}| = ℵo

Regards, WM