imagine

Real

But that is smaller than the naturals

Well now you just triggered a false vacuum decay on the far side of the galaxy. Way to go.

Now write a proof showing that your set is neither countably nor uncountably infinite and become the most famous mathematician I've replied to on Lemmy today

Sounds like you're asking the human brain to fire in a pattern it's not even wired for. Random noise in the web, or even definitionally impossible as "totally alien" might imply a configuration of neurons opposite of what we have. I feel like I'm having a hard time describing my thought here.

xighfkfutjgihugkghjgkckggdjjxubkctqjfhghhkhmhkhnvkcjfgrgshhgjdkguhjfjejtjgjffkcufjgjtiritu

Okay i did it, now what

the cardinality of a set is the number of things in it.

some sets have infinite items in them such as the counting numbers (there's always a bigger fish dot jpeg). but not all infinities are equal some are larger.

equality: if we can map a 1:1 rule between items in two sets with infinite items they are said to be equal infinities.

greater: but if we can map all in one set to another and note that there are still items left over, the first set has more things in it so if the other set has infinity items in it, this collection must have an even larger set of items in it, a greater tier of infinite.

a common example in math classes is mapping items in the real number between 0 and 1 to the counting numbers (1,2,3,...) using the rule 1>1/1, 2>1/2, 3>1/3,... we can see (0 to 1) has a 1:1 mapping but there are still more items (for instance 1/1.5). this shows there are more items in the real number line from 0 to 1 than there is items in the counting numbers. though both are infinite one infinity is larger.

so the meme. it's asking you to imagine a collection items that has greater number than the counting number infinity, but less than the next tier of infinity, those in the real number line. something which is hard to imagine because if it were easy we would have plugged that infinity tier into our tiering system.

You can project a 4D object onto a 3D space just like you can project a 3D object onto a 2D plane. If you use stereoscopic trickery you can for example watch a tesseract rotate on a phone screen. Don't ask me how I know but if you spend an evening doing that sorta thing on shrooms 4D geometry might start feeling intuitive to you. Your physical senses are limited to three dimensions, your mind genuinely isn't.

x ∈ ℝ⁴, there done

edit: if you want specifics, (1, 2, 3, 4)

Ok I did. Im just built different.

She did: https://en.wikipedia.org/wiki/Alicia_Boole_Stott

moves a cube

That actually sounds awesome. I'd pay to go to that show.

Just imagine them invariant to any 3D rotation.

Same as the cardinality of all reals. In fact, the cardinality of the set of all reals between 0 and 1 is the same as the cardinality of the set of all reals. https://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Sets_with_cardinality_of_the_continuum

Glad you made me look! I hadn't thought about whether there were sets with cardinality greater than the cardinality of the continuum. https://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Sets_with_greater_cardinality

Are you saying that there's nothing in between? Prove it, and turn modern mathematics inside out!

The problem is that rational numbers can be mapped (1 to 1) to the integers (e.g. just encode each rational number as an integer), so there are not more rational numbers than integers.

Well, there are more integers than naturals, yet both share the same cardinality. Also, I thing hilbert's hotel problem shows that rationals and naturals also share the same cartinality, somehow. You could arrange every rational in a line like the naturals and the integers.

But well tried, outstanding move.

That's not how cardinality works when dealing with infinite. For ex, there are the same number of prime number than number of integer. Yes, there are many non prime inter between 2 prime integer, but as long as you can "count" them, they have the same cardinality, which is called "aleph 0".

But you cannot "count" real number. There are actually more real between 0 and 1 than there are interger. This value is called "aleph 1".

Yes, there is also aleph 2, aleph 3,... There is not a single "infinite", but there are several one that don't have the same size.

Have a look to Hilbert's hotel paradox https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

If you could construct such a set, CH wouldn't be independent of ZFC

Olo is a good example. It's due to a quirk of human perception and the structure of our eyes. They basically designed a machine to try and stimulate the green detecting cones without stimulating the red detecting cones. Normally if something pure green hits your eyes, it stimulates those red cones too. So this is something our bodies are capable of perceiving but not something that we can ever perceive under normal circumstances.

Is it a "new color"? Not exactly. Did it take a good bit of imagination to conceive trying to get our brains to see it? Yes.