<h1 id="russells-paradoxes">Russell's Paradoxes<a aria-hidden="true" class="anchor-heading icon-link" href="#russells-paradoxes"></a></h1>
Book: <a href="/notes/49pcym4cka361wc6bo9qetc">The Man from the Future: The Visionary Life of John von Neumann</a>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>R be the set of all sets that are not members of themselves. If <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>R is not a member of itself, then its definition entails that it is a member of itself; if it is a member of itself, then it is not a member of itself.
<div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>R</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo>∉</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">R = \{x | x \not\in x\}</annotation></semantics></math>R={x∣x∈x}</div>
Then,
<div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>R</mi><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi>R</mi><mo>∉</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \in R \iff R \not\in R</annotation></semantics></math>R∈R⟺R∈R</div>

Russell's Paradoxes


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