무한의 세계

무한 집합의 크기Cardinality, 즉 원소의 개수를 수학에서는 '농도'라고 말한다. 유한 집합의 크기는 그대로 원소의 개수 이지만, 무한 집합의 경우는 원소의 개수를 낱낱이 셈하는 것은 불가능하기 때문에 '농도'라는 말을 사용하고 히브리 문자 ℵ로 표시하고 aleph라 읽는다.

자연수 전체의 개수, 즉 농도가 ℵ0인 집합은 \(\{ℵ0,ℵ1,ℵ2,…\}\) 와 같이 자연수로 번호Index를 붙일 수 있는 집합이라 하여 가부번 무한(집합)이라고 하고 \(\{ℵ_n : n ∈ ℕ\}\) 와 같다.

자연수 다음에 등장하는 것은 유리수(분수)이다. 분수는 직선 상에서 아무리 짧은 부분을 잡아도 그 속에 또 분수가 촘촘히 박혀 있다. 즉, 유리수의 집합은 조밀한 농도를 지닌다. 그러나, 분수 전체의 집합도 자연수의 집합과 1대 1 대응이 가능하며, 따라서 유리수 집합도 같은 가부번 집합이다.

그런데, 0 에서 1 사이의 선분 상의 점을 나타내는 수들의 집합은 아무리 해도 1, 2, 3, ...과 같이 번호를 매길 수 없다. 실수 집합의 농도는 연속의 선분 상의 점의 개수와 같기 때문에 연속체 농도 또는 비-가부번 농도라고 부르며, ℵ1으로 나타낸다. 즉, 실수의 집합은 자연수보다 훨씬 큰 무한 집합이다.

거시세계와 미시세계가 프랙탈 구조로서 연속되어 있다는 석가모니의 우주관에서 소름이 돋을수 밖에 없다! 우주의 무한 크기에 놀랄 것 없이 우리 자체가 훨씬 큰 무한일지 모르겠다.

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