Joana de Castro Arnaud<p><a href="https://mathstodon.xyz/tags/Tusky" class="mention hashtag" rel="tag">#<span>Tusky</span></a>, my <a href="https://mathstodon.xyz/tags/Mastodon" class="mention hashtag" rel="tag">#<span>Mastodon</span></a> <a href="https://mathstodon.xyz/tags/Client" class="mention hashtag" rel="tag">#<span>Client</span></a>, has no easy way to show what Hashtags I'm <a href="https://mathstodon.xyz/tags/Following" class="mention hashtag" rel="tag">#<span>Following</span></a>; I maintain a text file with them, but I got a better idea: posting them!</p><p>I'm <a href="https://mathstodon.xyz/tags/NowFollowing" class="mention hashtag" rel="tag">#<span>NowFollowing</span></a>: </p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a> <a href="https://mathstodon.xyz/tags/MathArt" class="mention hashtag" rel="tag">#<span>MathArt</span></a> <a href="https://mathstodon.xyz/tags/BeTheAlgorithm" class="mention hashtag" rel="tag">#<span>BeTheAlgorithm</span></a> <a href="https://mathstodon.xyz/tags/ToolsForMathematicalThought" class="mention hashtag" rel="tag">#<span>ToolsForMathematicalThought</span></a></p><p>Post your hashtags, plus <a href="https://mathstodon.xyz/tags/NowFollowing" class="mention hashtag" rel="tag">#<span>NowFollowing</span></a>, to share with us all.</p>
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#MathInduction
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The King<p><span class="h-card" translate="no"><a href="https://mathstodon.xyz/@mudri" class="u-url mention">@<span>mudri</span></a></span><br />Sort of. ZFC + "There is an inaccessible cardinal" implies you have a set that satisfies ZFC and is well founded.</p><p>It's more general and subtle than just that though. A measurable cardinal gives you a nontrivial elementary embedding of V into a well founded class, for example.</p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a></p>
The King<p>(7/7) Oh yeah, and optionally, you can help me with an experiment.</p><p>To help resolve the problem <span class="h-card" translate="no"><a href="https://mathstodon.xyz/@MartinEscardo" class="u-url mention">@<span>MartinEscardo</span></a></span> pointed out at <a href="https://mathstodon.xyz/@MartinEscardo/109552785927789702" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@MartinEscardo/</span><span class="invisible">109552785927789702</span></a>, I suggest the following:</p><p>- To follow this thread, follow the <a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a> tag (<a href="https://mathstodon.xyz/tags/Induction" class="mention hashtag" rel="tag">#<span>Induction</span></a> is a cooking tag)<br />- When posting in this thread, if you're trying out the experiment, add <a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a> to your post and make sure the visibility is public (not unlisted). You could include other tags that are relevant to your post too.</p><p><a href="https://mathstodon.xyz/tags/BeTheAlgorithm" class="mention hashtag" rel="tag">#<span>BeTheAlgorithm</span></a></p>
The King<p>(6/7) This is why the Burali-Forti paradox pops up in many different approaches to foundations (including type theory); if you can do induction over your own induction, you can often prove yourself consistent.</p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a></p>
The King<p>(5/7) Let's say you want to follow the simpler "PA is consistent because it is sound" route. Simply implementing the natural numbers isn't enough; you need to be able to define an inductive type for arithmetical sets as well! <a href="https://en.wikipedia.org/wiki/Arithmetical_set" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">en.wikipedia.org/wiki/Arithmet</span><span class="invisible">ical_set</span></a> In general for other arithmetics, you'll need to be able to do induction over the sets of things it inducts over. (For example, in full second order arithmetic you'd need induction over second order arithmetical sets of numbers.)</p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a></p>
The King<p>(4/7) As an example, consider the consistency of PA. This can be proven with epsilon_0 induction. But we don't need to think about Von Neumann ordinals. In type theory, being able to create an inductive type for the Cantor normal form (a tree like notation for ordinals below epsilon_0) is sufficient to prove PA consistent. <a href="https://en.wikipedia.org/wiki/Cantor_normal_form" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">en.wikipedia.org/wiki/Cantor_n</span><span class="invisible">ormal_form</span></a></p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a></p>
The King<p>(3/7)<br />- Large cardinal axioms in <a href="https://mathstodon.xyz/tags/SetTheory" class="mention hashtag" rel="tag">#<span>SetTheory</span></a> are often equivalent to asserting that there exists *well-founded* models with various properties, which gives you access to more induction.<br />- In Predicative Arithmetic by Edward Nelson (<a href="https://web.math.princeton.edu/~nelson/books.html" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">web.math.princeton.edu/~nelson</span><span class="invisible">/books.html</span></a>) it's shown that by severely weakening induction and then considering what numbers still satisfy inductive properties, you can get a system that might be acceptable in <a href="https://mathstodon.xyz/tags/ultrafinitism" class="mention hashtag" rel="tag">#<span>ultrafinitism</span></a>.</p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a></p>
The King<p>(2/7) Note that this induction is not just about Von Neumann ordinals. It takes different shapes in different foundations:</p><p>- In <a href="https://mathstodon.xyz/tags/arithmetic" class="mention hashtag" rel="tag">#<span>arithmetic</span></a> it is just natural number induction (although there are different forms of even this induction)<br />- In <a href="https://mathstodon.xyz/tags/TypeTheory" class="mention hashtag" rel="tag">#<span>TypeTheory</span></a>, induction is provided by inductive types. The stronger the type formers, the more induction you get.</p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a></p>
The King<p>(1/7) Mathematical induction is in some sense the most powerful mathematical technique.</p><p>I don't simply mean that it's really useful; I mean that meta-mathematically it often underpins the strength of a foundation. This is the main insight from ordinal analysis.</p><p>In addition, many famous theorems from "everyday" mathematics are logically equivalent to induction principles of various strengths. This is the main insight of reverse mathematics.</p><p><a href="https://mathstodon.xyz/tags/MathInduction" class="mention hashtag" rel="tag">#<span>MathInduction</span></a> <a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="tag">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="tag">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/ReverseMathematics" class="mention hashtag" rel="tag">#<span>ReverseMathematics</span></a></p>
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