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Okay, so I know, that the power set of any set $S$ has cardinality strictly greater, than cardinality of $S$. But the concept of power set is not that easy to visualize because it's more like an abstract construction.

For example, trying to visualize a power set of $\mathbb{N}$ one may think about the set of infinite binary sequences (each sequence is coding a subset of $\mathbb{N}$), but the simplest example to imagine of set greater than countable is $\mathbb{R}$ (or maybe a $[0, 1]$).

The question: is there an example of a set, that is simple to visualize and it has cardinality greater than continuum? Maybe, the set of all functions on $[0, 1]$, or maybe the set of all geometric shapes, i don't know?

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    $\begingroup$ For what it's worth, the set of all functions on $\mathbb R,$ or on $[0,1]$ has cardinality $2^c$ (provided the codomain has cardinality greater than one). The reasoning you gave for $\mathcal P(\mathbb N),$ gives a natural bijection between $\mathcal P(\mathbb R)$ and the set of all functions $\mathbb R\to \{0,1\}.$ Whether this space, or the set of all functions $\mathbb R\to \mathbb R$ is "easy to visualize" is a matter of taste, though. And, for instance, the set of all continuous functions $\mathbb R\to \mathbb R$ only has cardinality $c.$ $\endgroup$ – spaceisdarkgreen Jul 30 '18 at 23:04
  • $\begingroup$ Thank you, i didn't think about power set as of a set of all functions to $\{0, 1\}$. I think, your example completely answers my question. $\endgroup$ – grentank Jul 30 '18 at 23:10
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There is an inherent problem with your question. Actually two.

The first is that visualizing something requires some sort of inherent structure. We can visualize the real numbers as a line, or the natural numbers as dots, and $\Bbb R^3$ as the room you are sitting in.

When you talk about cardinality, you—by definition—omit the structure. Note that $\Bbb N$ and $\Bbb Z$ are very different, but they are both countable. Even $\Bbb N$ with the usual order, and with the divisibility relation ($m\mid n\iff\exists k\in\Bbb N:mk=n$) are very different visually, despite being two structures of the same set.

So can we come up with an easy to visualize structure? Sure. We have linear orders of any cardinality, as follows from the axiom of choice, so we can take one which "locally" looks like $\Bbb R$, or even like $\Bbb N$ (again, locally is a key word here). Does that mean something to you? How is that different from $\Bbb R^2$ viewed with the lexicographic order? It really doesn't.

The second problem is relying on "visualizing". Yes, you can visualize the function $f(x)=x$ or $f(x)=e^x$. If you try very hard, you can even differentiate in your mind's eye between $\ln x$ and $\sqrt x$ (but I can't, their graphs are just too similar). Mathematics is built on definitions. You could argue, and correctly so, that $\Bbb Q$ and $\Bbb{R\setminus Q}$ are extremely different objects. While both are linearly ordered metric spaces, only one of them is completely metrizable, only one of them is a $G_\delta$ subset of $\Bbb R$, only one of them is uncountable, and only one of them is a field. Even though, they are both totally disconnected and without isolated points.

So if you think about either $\Bbb Q$ or $\Bbb{R\setminus Q}$ visually, you are likely to think about a line with "missing dots almost everywhere", but also dots are still everywhere". Sort of to signify that both the set and its complement in the real line are dense and co-dense sets. And therein lies the rub. We just talked about how different these mathematical objects are, from several important and distinctive perspectives. Yet, visually speaking, they "look and feel the same".

This is why visualizing is not a good direction. It is not a good motivating tool. Abstract mathematics, and set theory in particular, is prone to confuse you if you try to visualize things as a crutch and a scaffolding device. You should rely on the definitions you have, and work carefully. With time you will gain the proper intuition for handling definitions related to infinite sets and their cardinality. It's not the easy way, but you won't be cheating yourself with the idea that you can somehow visualize well enough to make the distinction between two objects which are very different from one another.

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