- #1

Quasimodo

- 45

- 8

- TL;DR Summary
- The Continuum Hypothesis and Number e

**Summary:**The Continuum Hypothesis and Number e

Now, I must ask a very stupid question:

When taking: $$\lim_{_{n \to \infty} } (1+\frac{1}{n})^n=e\\$$ the ##n## we use take its values from the set: ## \left\{ 1,2,3 ... \right\} ## which has cardinality ## \aleph_0 ##, which is equivalent maybe, I say maybe to writing: $$\ (1+\frac{1}{\aleph_0})^{\aleph_0}=e\\$$

Upon: $$\lim_{_{n \to \infty} } (1+\frac{1}{n})^{2n}=e^2\\$$ we take, $$\ (1+\frac{1}{\aleph_0})^{2\aleph_0}=e^2\\$$

So, since two equal power bases give two different results, we have to assume that their exponents are different hence: $$ 2\aleph_0 > \aleph_0 $$