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set theory – Can be this “handwaving” idea about “counting” reals somehow put on solid ground?

We know that the Cantor’s cardinality of a countable set is $aleph_0$ and the cardinality of continuum is $2^{aleph_0}=aleph_0^{aleph_0}$. Unfortunately, this measure is based on the idea of bijection, so sets of different volume can have the same cardinalities.

On the other hand, there is an idea of “numerocities” (which requires ordered or metric space), linked to the divergent integrals and series. For a subset of non-negative integers $S$ the numerocity is represented by a generally divergent series:

$$sum_{k=0}^infty p_S(k)$$

where $p_S(k)$ is the membership function, equal to $1$ if $kin S$ and $0$ otherwise. It allows to compare (using properties of divergent series) such sets as even and odd numbers, prime numbers, etc, giving more precise notion of the set’s size than cardinality.

The question is, can we somehow introduce a concept similar to numerocity to the uncountable sets, in such a way that it would reflect the volume of those sets? In other words, the set $(0,2)$ should have twice he numerocity of $(0,1)$. This is desirable…

First, let us agree on some symbols. Let use $omega_-=sum_{k=1}^infty 1$ for numerocity of natural numbers, and $omega_+=sum_{k=0}^infty 1=omega_-+1$ for numerocity of non-negative integers.

The following is pure handwaving, please don’t beat me hard.

First, let us consider the binary representation of the reals from $(0,1)$:
$0.1101001…$. Here each digit can be either $0$ or $1$. Since we start from position $1$ and have positions corresponding to all natural numbers, we can say we have $omega_-$ positions. So, the whole numerocity is $2^{omega_-}$.
Of course, if we take another base rather than $2$, we will have different expression for that numerocity. But this should not be confusing because numerocity is often dependent on ordering and filtering. So, if we make the same infinite set more dense, its numerocity changes (grows).

Can we somehow establish the expression for numerocity of reals on an interval, that would not depend on the chosen base for representation?

As I see it, the only way to avoid dependence on the base is to allow the base to go to infinity. So, we transcend to a representation in an infinite base. The digits start from $0$ but can be any natural number. As such (because we start from $0$), we have $omega_+$ digits. And the whole numerocity is $omega_+^{omega_-}$. Please don’t ask me how a particular number would be represented in such system. One can think of it as of some kind of limit. Also, since we do not have the biggest digit, the number $1$ does not belong here, so this numerocity represents the range $(0,1)$ rather than $(0,1)$.
I would point out that the expression of the form $omega_+^{omega_-}$, $omega_-^{omega_-}$ etc, often appear in the theory of divergent series, so this is not something “unseen”. It also fits well with the similar expression for the cardinality of continuum, which is a similar but less refined measure.
The numerocity of the whole real line then would be expressed as $(omega_++omega_-)omega_+^{omega_-}$.

I understand, these are very loose deas, particularly the idea of “infinite base” for a digital representation is unjustified. But I wonder whether some grains of it can be put on solid ground.

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The distinction between function and the class in the 1st principle of the SOLID design principle

So, I have been trying to learn about the SOLID principle and want to integrate the knowledge into my daily work.

But, the way we are taught about the class and the OOP, is that we should create an object which behaves like real-world objects. But this directly contrasts with the 1st principle of the SOLID principle which states that “A class should have one and only one reason to change, meaning that a class should have only one job.”

But it’s not what the real object remotely behaves like, in real life, an object is able to do multiple things and has many reasons to change. That’s why when we create the class we write all the properties and methods which can change the behavior of the object.

And one other problem with the 1st Principle is that, let’s suppose we fully apply the 1st principle then we end up creating a lot of classes and then this blurs the lines of what is a function and what is class. All the functions end up being the class and then I don’t know what are the jobs of the classes and the function.

What is missing here? How should we approach this?

sharepoint online – Differences between “Green tick icons” & “Solid green circle with the white check mark” in OneDrive status column

The green tick icon means it is locally available (stored on your device) and can be opened when offline but if you have Storage Sense enabled these files will become online only again after an amount of time that you have set.

The solid green circle with a white tick means the file will always will be available on your device if you are offline (stored locally). The Storage Sense policy would not make this file online only after a certain period of time.

More information can be found here:

https://support.microsoft.com/en-us/office/save-disk-space-with-onedrive-files-on-demand-for-windows-10-0e6860d3-d9f3-4971-b321-7092438fb38e

dnd 5e – Turning mud to solid ground

What cantrip, if any, would be able to turn mud into solid ground? Would mold earth work? Would shape water?

Context: I’m making a homebrew race who live in an area of bog, and I thought it would be interesting to have them making their own homes with racial spellcasting.

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Why is a solid, scalable, and modular infrastructure important for agile software development

Why is a solid, scalable, and modular infrastructure important for agile software
development

This is a calculus 3question referring to finding the total mass of the solid in kilograms [closed]

These two questions are put together and the first one will be needed to be done to be worked upon on the second one .

solid color to sprite (javascript game)

Hi I recently created a javascript game, in this game, math random is used to pick a random shape.

The shapes are in strings, and the associated numbers within the string are then used to show a color.

Is there any way to replace the simple hex colors with an image?

For example:

else if (shape === "H"){
    return ((8,8,8),
            (8,8,8),
            (8,8,8));

The “8” ties into the following string to get the #F10B38 color.

const color = (null,"#FF2D00","#FF9300","#51FF00","#00FF93","#0087FF","#4E49A7","#9649A7","#F10B38");