<p><strong>Foundations</strong></p>
<ul>
<li><a href="https://math.stackexchange.com/a/1802028/21820">Axioms</a> &amp; <a href="https://math.stackexchange.com/a/1399880/21820">Definitions</a></li>
<li><a href="https://math.stackexchange.com/a/1668149/21820">Contradiction</a>
&amp; <a href="https://math.stackexchange.com/a/3699299/21820">Game semantics</a>
&amp; <a href="https://matheducators.stackexchange.com/a/10034/1550">True induction</a></li>
<li><a href="https://math.stackexchange.com/a/1334753/21820">Circularity in mathematics</a>
&amp; <a href="https://math.stackexchange.com/a/1888389/21820">'Paradoxes' resolved</a></li>
<li><a href="https://math.stackexchange.com/a/1808558/21820">Building blocks</a>
&amp; <a href="https://math.stackexchange.com/q/2486348/21820">Incompleteness theorems</a></li>
<li><a href="https://math.stackexchange.com/a/1684208/21820">Logic texts</a></li>
</ul>
<p><strong>Natural deduction</strong></p>
<ul>
<li><a href="https://math.stackexchange.com/a/1684204/21820">Rules</a>
&amp; <a href="https://math.stackexchange.com/a/1788516/21820">Examples</a></li>
</ul>
<p><strong>Rigour + Intuition</strong></p>
<ul>
<li><a href="https://math.stackexchange.com/a/2978757/21820">Naturals</a></li>
<li><a href="https://math.stackexchange.com/a/1847519/21820">Reals</a>
&amp; <a href="https://math.stackexchange.com/a/1790356/21820">Real exponentiation</a>
&amp; <a href="https://math.stackexchange.com/a/2360276/21820"><span class="math-container">$\cos,\sin$</span> &amp; <span class="math-container">$\exp(iπ)+1=0$</span></a></li>
<li><a href="https://math.stackexchange.com/a/2118909/21820">General differentiation</a>
&amp; <a href="https://math.stackexchange.com/a/2176223/21820">applications</a></li>
<li><a href="https://math.stackexchange.com/a/1698898/21820">Matrix multiplication</a></li>
</ul>
<p><strong>Beautiful stuff</strong></p>
<p>Puzzles:</p>
<ul>
<li><a href="https://math.stackexchange.com/q/1096592/21820">Rubik's cube &amp; other permutation puzzles</a></li>
</ul>
<p>Finite difference:</p>
<ul>
<li><a href="https://math.stackexchange.com/a/2319671/21820">Indefinite summation (e.g. sum of cubes)</a></li>
<li><a href="https://math.stackexchange.com/a/1414669/21820">Linear recursive sequences</a>
&amp; <a href="https://math.stackexchange.com/posts/comments/2749264">their indefinite sum</a></li>
</ul>
<p>Asymptotic expansion:</p>
<ul>
<li><a href="https://math.stackexchange.com/a/1298332/21820">k-times-differentiable function</a></li>
<li><a href="https://matheducators.stackexchange.com/a/7280/1550">Central difference formulae</a></li>
<li><a href="https://math.stackexchange.com/a/800070/21820">Newton-Raphson convergence criteria</a></li>
<li><a href="https://math.stackexchange.com/a/2504179/21820">Systematic computation of limits</a></li>
</ul>
<p>Smoothing:</p>
<ul>
<li>To prove AM-GM &amp; Jensen's inequality</li>
<li><a href="https://math.stackexchange.com/a/868065/21820">Maximum area of polygon given perimeter</a></li>
<li><a href="https://math.stackexchange.com/a/1357293/21820">Maximum expression value using brackets</a></li>
</ul>
<p>Multi-dimensional analysis:</p>
<ul>
<li><a href="https://math.stackexchange.com/a/95867">Multi-dimensional IVT</a>, which proves:</li>
<li>Poincaré–Miranda theorem</li>
<li>Ham sandwich theorem [consider all hyperplanes that bisect one piece]</li>
</ul>
<p>Inversion:</p>
<ul>
<li><a href="https://math.stackexchange.com/a/625571/21820">Apollonius' Problem</a></li>
</ul>
<p><strong>Common oversights</strong></p>
<ul>
<li><a href="https://matheducators.stackexchange.com/a/7984/1550">Separating variables</a></li>
<li><a href="https://math.stackexchange.com/a/1293675/21820">'One-step' analysis to find expectation</a></li>
<li><a href="https://matheducators.stackexchange.com/a/10033/1550">Explanation of induction</a></li>
</ul>

user21820