night mesa
this is euclidean space in mathematics. r^4, r^100, whatever.
let's say you are a superpowered being. you have the magical power to destroy points each point would require 1 power from you. destroying infinite points would require of you infinite power.
does any r^m space have more points (thus more power needed to destroy) than r^n? is r^10000 harder to destroy than r^30? it would all depend on how many points there are, right?
hollow holly
this is euclidean space in mathematics. r^4, r^100, whatever.
let's say you ar...
Gemini: This is a fascinating question that delves into the nature of infinity and the concept of "more" when dealing with infinite sets.
The Paradox of Infinity
In mathematics, particularly when dealing with infinite sets, our intuition often fails us. One of the most counterintuitive aspects of infinity is that different infinities can have the same cardinality, meaning they can be paired up one-to-one.
The Case of Euclidean Spaces
While it might seem intuitive that a higher-dimensional space like R^10000 should have "more" points than a lower-dimensional space like R^30, this isn't necessarily the case. In fact, both of these spaces have the same cardinality, which is the cardinality of the set of real numbers, often denoted as 𝔠 (pronounced "c").
The Power of the Continuum
The cardinality 𝔠 is often referred to as the "power of the continuum." It's the cardinality of the set of real numbers, which is also the cardinality of the set of points in any Euclidean space of finite dimension.
Conclusion
So, even though R^10000 might seem vastly larger than R^30, they both contain the same "number" of points, and thus, as a superpowered being, you would require the same infinite amount of power to destroy all the points in either space.
This surprising result highlights the counterintuitive nature of infinity and the limitations of our intuitive understanding when dealing with such concepts.
night mesa
thanks
hollow holly
How can we challenge these ideas? I think the key is in the last paragraph.
This surprising result highlights the counterintuitive nature of infinity and the limitations of our intuitive understanding when dealing with such concepts.
night mesa
@turbid pumice you wanted to take a go at this
turbid pumice