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Okay, I think that jogged my memory. So, the idea is that no matter what mathematical mapping you create from Natural -> Real, it will always be possible to fi
by jpcfl 4y ago
Okay, I think that jogged my memory.
So, the idea is that no matter what mathematical mapping you create from Natural -> Real, it will always be possible to find a Real number that is not in that mapping. Therefore, |Real| > |Natural|.
Still, it feels wrong. If my "function" is an immortal monkey I've trained to grab one item from an unlimited Natural number bucket, and another from an unlimited Real number bucket, and put them together, that process could go on ad infinitum. It's hard to imagine one bucket as been larger than the other, when both are endless.
- hobs 4y agoIf you had a bucket with 10 million options in one and 100m in another, you'd take 10 million picks before you realized via a 1-1 picking that one was smaller than the other. Infinities obviously dont collapse and "end", but seeing infinities "grow" faster than others makes a lot of sense to me.
- vishnugupta 4y agoInfinity as a concept is not easy to reason about because it’s not something we ever encounter in our daily lives. It leads to interesting paradoxes such as Hilbert Hotel. Coming to Natural and Real numbers, one way I’d like to think is that real numbers are more “dense” (or continuous) compare to integers which are discrete. A cup full of sugar and cup full of water could be of equal volume but you can count sugar grains and not water because water is not grainy and is “continuous”.