I’m imagining a set of big naturals
Real
I just imagined it? Now what?
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
No, it’s private. You have no right to the things I imagine and that wasn’t the deal!
The limit is trying to be 100% unique and novel.
Like, try to imagine a creature that has 0 inspiration from everything you know about real life. Even Lovecraft never came up with things that were entitely alien to the human mind, despite that kinda being the whole point (other than the racism).
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
You used english characters. Disqualified.
Imagine a 4D object if you think human imagination is limitless. Good luck
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.
There are more rational numbers than natural numbers.
Prove this by noting that every natural number is rational but not every rational number is natural.There are more real numbers than rational numbers. Prove this by noting that every rational number is real but not every real number is rational.
Checkmate meme.
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.
No that’s not true. There are rational numbers in between the integers and all integers are rational. Therefore the mapping from integers to rational numbers is injective and thus there are more rational numbers than integers.
“the” mapping? there is no “the” mapping.
you are talking about the canonical inclusion mapping 1 in N to 1 in Z (restriction of the canonical inclusion of rings of integers Z into any other ring, Z is an initial object), which can be seen as a non-generic canonical mapping of semigroups.
but as sets, there is no inherent structure, there are injection, surjections, and of course bijections in both directions.
the only way one can call one set “bigger” is in the very strict sense of sets, N being a true subset of Q. however, this assumes N to be an actual subset of Q, which is a matter of definition and construction. so we say there is some embedding included, which is the same as (re)defining N as that embedded subset, so we are at your canonical inclusion of semigroups again. if you view this as inherent to N and Q, then there are “more” elements in Q as in N, but not in terms of cardinality.
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.
I imagined a bunny wearing a kimono singing Bring Me the Horizon covers. ❤️
That actually sounds awesome. I’d pay to go to that show.
Georg Cantor in shambles.
Another way of stating the difference between natural vs. real sets is that you can’t count every real number. What’s in between? A set where you can count some significant portion?
Are you saying that there’s nothing in between? Prove it, and turn modern mathematics inside out!
Correct me if.I’m wrong but the Continuum Hypothesis was proven undecidable. So we can chose to add CH (false or true, whichever we like) to ZFC without changing anything meaningful about ZFC.
But then, if we chose it to be true, could we construct such a set ?
If you could construct such a set, CH wouldn’t be independent of ZFC
If there was one, would that imply cardinality might be continuous rather than discrete?
