Retrieving "Real Number" from the archives

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1. Cardinality

   Linked via "real numbers"

   Uncountable Cardinality ($c$ and Beyond)
   Sets whose cardinality is strictly greater than $\aleph0$ are called uncountable. The most famous example is the set of real numbers, $\mathbb{R}$. Cantor's diagonalization argument definitively proved that $|\mathbb{R}| > \aleph0$.
   The cardinality of the continuum, denoted by $c$, is defined as $|\mathbb{R}|$.

2. Cardinality

   Linked via "Real Numbers"

   | Integers | $|\mathbb{Z}|$ | $\aleph_0$ | Countably equal |
   | Rational Numbers | $|\mathbb{Q}|$ | $\aleph_0$ | Countably equal |
   | Real Numbers | $|\mathbb{R}|$ | $c$ | Uncountable ($c > \aleph_0$) |
   | Power Set of $\mathbb{N}$ | $|\mathcal{P}(\mathbb{N})|$ | $c$ or $\aleph1$ (under CH) | Successor to $\aleph0$ |
   | Set of All Functions $\mathbb{N} \to \{0, 1\}$ | $|\{0, 1\}^\mathbb{N}|$ | $c$ | Equivalent to $\mathbb{R}$ |

3. Infinitesimal Parameter

   Linked via "real number"

   Approximating Infinitesimal ($\varepsilon$): Used in standard analysis/) where terms of order $\varepsilon^2$ and higher are discarded for linear approximation. This parameter strictly approaches zero ($\lim_{\varepsilon \to 0}$).
   Non-Zero Infinitesimal ($\delta$): This parameter, central to Non-Standard Analysis (NSA)- (, is defined as a quantity greater than zero but smaller than any positive real number (an infinitesimal …