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1. Errata for Proof Theory by Gaisi Tak... Errata for Proof Theory by Gaisi Takeuti

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   # Errata for Proof Theory by Gaisi Takeuti

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   These are possible errors (mostly typos) I have found in the Dover republication of *Proof Theory* by Gaisi Takeuti.

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   > This Dover edition, first published in 2013, is an unabridged republication of the work originally published in 1987 by North-Holland, Amsterdam, as Volume 81 in the series *Studies in Logic and the Foundations of Mathematics*. The first edition was published in 1975. 

2. Takeuti's notion of indication and f... Takeuti's notion of indication and full indication

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   # Indication and full indication 

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   In Gaisi Takeuti's Proof Theory, he introduces the notion of **substitution/replacement** as follows:

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   > **Definition 1.4.** Let $A$ be an expression, let $\tau_1, \dots, \tau_n$ be distinct primitive symbols, and let $\sigma_1, \dots, \sigma_n$ be any symbols. By  

3. Algebraic Topology in the horror mov... Algebraic Topology in the horror movie Ring (1998)

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   # Algebraic Topology in the horror movie Ring (1998)

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   In the 1998 horror movie [Ring (リング)](https://en.wikipedia.org/wiki/Ring_(film)), the protagonist's ex-husband happens to be a mathematics professor named Takayama Ryūji (高山 竜司). He is played by [Sanada Hiroyuki (真田 広之)](https://en.wikipedia.org/wiki/Hiroyuki_Sanada) known for his music and roles in Hollywood action movies such as [The Last Samurai](https://en.wikipedia.org/wiki/The_Last_Samurai) and [John Wick: Chapter 4](https://en.wikipedia.org/wiki/John_Wick:_Chapter_4). He is caught by the vengeful ghost [Sadako (貞子)](https://en.wikipedia.org/wiki/Sadako_Yamamura) doing some mathematics (presumably some Algebraic Topology) and is mysteriously murdered ([scene on YouTube](https://www.youtube.com/watch?v=nFIk45uH2qo)). Throughout the movie there are several scenes which features the characters mathematics. Some of his books contain some [Ring theory](https://en.wikipedia.org/wiki/Ring_theory), however, most of his books pertain to Topology or Physics.

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   The following are some rough timestamps and brief descriptions of the mathematics in the scene: