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julia 🦄 @hypv.bsky.social · 4h

smh helen don't you know the catholic church is epic and leftist now

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julia 🦄 @hypv.bsky.social · 6h

omega male just means you can get estrus and pregnant

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julia 🦄 @hypv.bsky.social · 6h

is an omega-parody and also an ultrafinitist at the same time

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julia 🦄 @hypv.bsky.social · 6h

imagine how horrifying it'd be to call on your fiance late on a stormy night and you discover he's been cheating on you creating life with his assistant fritz, and then he makes you watch

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Reposted by julia 🦄

Sophie Sherry @sophiesherry.bsky.social · 10h

During a military-style raid on a building in Chicago’s South Shore, one resident heard a knock on his door.   It wasn’t the feds — it was a mom and her 7-year-old daughter, pleading for help.

He let them hide in his unit for the next 3 days.

chicago.suntimes.com/immigration/...

Neighbor shielded 7-year-old during South Shore federal raid: ‘I didn’t want them to take her’

During the Sept. 30 raid one tenant protected a terrified girl and her mom. Remnants at the complex, including a detailed map of all the units, offer clues to what authorities may have known before th...

chicago.suntimes.com

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julia 🦄 @hypv.bsky.social · 8h

we don't talk enough about what a tough economy this is for bulls. who has the disposable income anymore to pay rent, go on lots of dates, and also buy her husband a switch 2?

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julia 🦄 @hypv.bsky.social · 9h

Liz Warren lmao

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julia 🦄 @hypv.bsky.social · 9h

yes, adding more structure allows you to make more distinctions between elements

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julia 🦄 @hypv.bsky.social · 9h

take a transcendence basis of C over Q, use zorn's lemma on the basis, sswapping elements around, to produce as automorphism of C

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julia 🦄 @hypv.bsky.social · 10h

look at this little gay boy in his collar

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julia 🦄 @hypv.bsky.social · 10h

but how we know it's two things? maybe it's only one thing

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julia 🦄 @hypv.bsky.social · 10h

when the book club meeting moves to everyone talking about how a different book is so much better than the book we got together to talk about

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julia 🦄 @hypv.bsky.social · 12h

technology by rapists for rapists

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julia 🦄 @hypv.bsky.social · 12h

just saw an ad for a "AI meets XR" app and the first example in their video of how you can use generative AI with your meta glasses was visually undressing a woman to her underwear

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julia 🦄 @hypv.bsky.social · 12h

that's when honolulu does it, it's a great time of the year for it

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julia 🦄 @hypv.bsky.social · 13h

*2^{2^{\aleph_0}} is clearly what i meant. sorry

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julia 🦄 @hypv.bsky.social · 13h

*you need more than a basis as a vector space, you need a transcendence basis of C over Q

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julia 🦄 @hypv.bsky.social · 13h

the wild automorphisms are because C is a Q-vector space. so you can well-order a hamel basis and recursively construct an automorphism from it (the same trick doesn't work with R because the order of R can be recovered from its algebra)

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julia 🦄 @hypv.bsky.social · 13h

based and r/atheism-pilled to have mother teresa be a go-to example here

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julia 🦄 @hypv.bsky.social · 14h

no. you're a lizard, not a snake

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julia 🦄 @hypv.bsky.social · 14h

@bitdizzy.bsky.social i was inspired by your posting about why equality is hard

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julia 🦄 @hypv.bsky.social · 14h

A lot of people think that the complex numbers are more nicely behaved than the real numbers. But in fact ℝ is better than ℂ, and model theory and Gottfried Wilhelm von Leibniz can help us to understand why. Part 2/2 #MathSky

$\begin{frame}{Leibnizian structures} A structure $\Mcal$ is \highlightb{Leibnizian} if it satisfies Leibniz's identity of indiscernibles. \begin{itemize} \item In model theoretic language, if for all $x,y \in \Mcal$, we have $x = y$ iff $\type(x) = \type(y)$, \item Where $\type(x) = \{ \varphi : \Mcal \models \varphi(x) \}$ is the \highlightb{type} of $x$. \end{itemize} \bigskip \highlightv{Example:} $\mathbb Q$ is Leibnizian because e.g. $3/2$ is only number satisfying the property of being $1 + 1 + 1$ times the multiplicative inverse of $1 + 1$. \bigskip \highlightv{Example:} But if we consider $\mathbb Q$ merely as an order, not as an algebraic structure, then it is \highlightr{not} Leibnizian. (Proof: Exercise.) \bigskip \highlightv{Silly Example:} Any structure can be made Leibnizian by adding names for every object to the language, because then ``I am $x$'' is in the type of $x$. \end{frame}$ $\begin{frame}{$\Rbb$ is good} \highlightv{\textbf{Theorem} (Dedekind)\textbf{:}} $\Rbb$, considered as an algebraic structure, is Leibnizian. \begin{itemize} \item Every rational number is uniquely definable, so properties like ``$3/2 < x$'' appear in the type of $x$. \item You can express $<$ using the fact that nonnegative numbers don't have square roots. \item Thus, $\type(x)$ includes all the information about the \highlightb{Dedekind cut} identifying $x$. \item Real numbers are identical iff they have the same Dedekind cut. \item For example, $0.999\ldots$ and $1$ are identical because for any rational number $q$, we have $q < 0.999\ldots$ iff $q < 1$. \end{itemize} \end{frame}$ $\begin{frame}{Automorphisms} \textbf{\highlightv{Fundamental Theorem of Isomorphisms:}} If $\pi : \Mcal \cong \Ncal$ is an isomorphism, then for any property $\varphi$ and any $x \in \Mcal$, \[ \Mcal \models \varphi(x) \iff \Ncal \models \varphi(\pi x). \] In other words, $\type(x) = \type(\pi x)$ for every $x \in \Mcal$. \bigskip % \textit{\highlightv{Corollary:}} If $\Mcal$ has a nontrivial automorphism then it is not Leibnizian. \begin{itemize} \item Because $x$ and $\pi x$ have the same type. \end{itemize} \bigskip \textit{\highlightv{Remark:}} \highlightb{Rigid} ($=$ no nontrivial automorphisms) doesn't imply Leibnizian. \begin{itemize} \item If the language of $\Mcal$ has $\kappa$ many properties then $\Mcal$ has $2^\kappa$ many different possible types. \item Thus, Leibnizian $\Mcal$ can have cardinality at most $2^\kappa$. \item But there are rigid structures of any cardinality, e.g. well-orders. \end{itemize} \end{frame}$ $\begin{frame}{$\Cbb$ is bad} \textbf{\highlightv{Theorem:}} $\Cbb$ has a nontrivial automorphism, so it is not Leibneizian. \begin{itemize} \item One automorphism is the one which flips the complex plane vertically. \item Assuming the \highlightv{axiom of choice} there are $2^{\aleph_0}$ many other automorphisms, but they are hard to write down explicitly. \end{itemize} \end{frame}$

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julia 🦄 @hypv.bsky.social · 14h

A lot of people think that the complex numbers are more nicely behaved than the real numbers. But in fact ℝ is better than ℂ, and model theory and Gottfried Wilhelm von Leibniz can help us to understand why. Part 1/2 #MathSky

$\title{Why $\mathbb R > \mathbb C$, and how model theory can help explain why} \author{Julia Redacted} \begin{document} \begin{frame} \titlepage \end{frame}$ $\begin{frame}{How do you tell when two things are different?} Mathematicians love to gloss over this, but it's \highlightv{really hard} to tell whether two things are equal. One object can be presented in different ways and it can be non-trivial to see they give the same thing. \bigskip \begin{itemize} \item If you were ever confused why $1 = 0.999\ldots$ you've experienced this. \end{itemize} \bigskip If you're interested in some mathematical structure \highlightb{$\Mcal$} consisting of objects, how do you tell them apart? \end{frame}$ $\begin{frame}{Leibniz's Identity of Indiscernibles} \begin{columns} \begin{column}{0.2\textwidth} \includegraphics[width=0.75in]{leibniz.jpg} \end{column} \begin{column}{0.8\textwidth} \highlightp{\textbf{\Large``}}Two objects are identical iff they have the same properties.\highlightp{\textbf{\Large''}} \vspace{.5in} \end{column} \end{columns} \begin{columns} \begin{column}{0.8\textwidth} \ Mathematically: $x$ and $y$ are objects in $\Mcal$. Then $x = y$ iff $\Mcal \models \phi(x) \iff \Mcal \models \phi(y)$ for every property $\phi$. \begin{itemize} \item $\Mcal \models \varphi(x)$ is model theorist speak for \highlightv{$x$ has property $\varphi$ in the structure $\Mcal$}. \item It's circular to have a property like ``identical to $x$''. We only mean properties you can express in the \highlightv{language} of the structure, e.g. the language of rings. \end{itemize} \end{column} \end{columns} \end{frame}$ $\begin{frame}{Telling numbers apart} \begin{itemize} \item You can distinguish $1$ from any other number, because $1$ is the only number which is a multiplicative identity. \item You can distinguish $\pi - e$ from $\pi/e$ because one is negative and the other is positive. \item You might think you can distinguish $i$ and $-i$ because only one is on the top half of the complex plane, but what if you are holding $\Cbb$ upside down? \end{itemize} \end{frame}$

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julia 🦄 @hypv.bsky.social · 18h

sorry for making a basic bitch feminist point, but the way this carefully workshopped statement invokes the family to signal higher moral valence

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julia 🦄 @hypv.bsky.social · 19h

opossumgirl who posts about being roadkillbait

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