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<body>
<div id="content" class="content">
<h1 class="title">uni uni uni blog blog blog</h1>
<div id="table-of-contents" role="doc-toc">
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#orgc7181d4">1. reasons for 2 to not exist</a>
<ul>
<li><a href="#org65779f8">1.1. uni what the fuck are you talking about</a></li>
<li><a href="#orgc7f40b1">1.2. set-theoretical stuff</a>
<ul>
<li><a href="#org3e2b8ba">1.2.1. regularity</a></li>
</ul>
</li>
<li><a href="#org897cb1b">1.3. categories</a>
<ul>
<li><a href="#orgfdce46d">1.3.1. 2 is cartesian closed</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-orgc7181d4" class="outline-2">
<h2 id="orgc7181d4"><span class="section-number-2">1.</span> reasons for 2 to not exist</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org65779f8" class="outline-3">
<h3 id="org65779f8"><span class="section-number-3">1.1.</span> uni what the fuck are you talking about</h3>
<div class="outline-text-3" id="text-1-1">
<p>
In mathematics, there’s a bunch of ways infinities can be large. Usually, the existence of these
can not be derived <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">from the usual axioms</a> yielding an important logical component to the study
of these <a href="https://en.wikipedia.org/wiki/Large_cardinal">large cardinals</a>. But many of these properties include the assumption that the
cardinal number in question is infinite - once we remove that assumption (or sometimes
restate the property entirely) we can see that 2 is, in many regards, a pretty huge
number.
</p>
</div>
</div>
<div id="outline-container-orgc7f40b1" class="outline-3">
<h3 id="orgc7f40b1"><span class="section-number-3">1.2.</span> set-theoretical stuff</h3>
<div class="outline-text-3" id="text-1-2">
</div>
<div id="outline-container-org3e2b8ba" class="outline-4">
<h4 id="org3e2b8ba"><span class="section-number-4">1.2.1.</span> regularity</h4>
<div class="outline-text-4" id="text-1-2-1">
<p>
A <a href="https://en.wikipedia.org/wiki/Cardinal_number">cardinal number</a> \(\kappa\) is <a href="https://en.wikipedia.org/wiki/Regular_cardinal">regular</a> if it cannot be expressed as an indexed sum of numbers smaller
than \(\kappa\) where the indexing number is also smaller than \(\kappa\), i.e. there’s no index
\(\iota < \kappa\) such that a family of sets \(A_i\) for \(i \in \iota\) exists for which
\(\sum_{i \in \iota} A_i > \kappa\) holds.
</p>
<p>
For example, the cardinal \(\aleph_0\), as in the size of the natural numbers, is regular,
since every finitely indexed sum of finite numbers is finite.
However, the only finite regular cardinals are 0, 1 and 2 - every other natural number \(n\) can be
expressed as the sum \(1 + (n-1)\), both of which are smaller than n.
</p>
<p>
Since regularity is an important part of the definition of <a href="https://en.wikipedia.org/wiki/Inaccessible_cardinal">inaccessible cardinals</a>, which are
indeed quite large, I guess this kind of makes 2 way larger in relationship to 0 and 1 at least.
</p>
</div>
</div>
</div>
<div id="outline-container-org897cb1b" class="outline-3">
<h3 id="org897cb1b"><span class="section-number-3">1.3.</span> categories</h3>
<div class="outline-text-3" id="text-1-3">
</div>
<div id="outline-container-orgfdce46d" class="outline-4">
<h4 id="orgfdce46d"><span class="section-number-4">1.3.1.</span> 2 is cartesian closed</h4>
<div class="outline-text-4" id="text-1-3-1">
<p>
Consider the category with two objects and one arrow between them, e.g. the <a href="https://ncatlab.org/nlab/show/interval+category">interval category</a>.
It has all limits and is cartesian closed, a property commonly associated with things that are rather big, e.g. the category of sets
</p>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: uni</p>
<p class="date">Created: 2024-02-25 So 12:31</p>
</div>
</body>
</html>