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\catcode\^^J=10

\def\Id{\mathbf{Id}}

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\footnotesize

Logics of Worlds: Being and Event, 2'' (2006, translation 2009):

\url{https://www.bloomsbury.com/uk/logics-of-worlds-9781441172969/}

\ssk

These notes are at:

\ssk

See:

\url{http://angg.twu.net/LATEX/2020favorite-conventions.pdf}

\url{http://angg.twu.net/math-b.html\#favorite-conventions}

I wrote these notes mostly to test if the conventions above
are good enough.

}

% (find-badioulowpage (+ 18 153) "II.3. Algebra of the Transcendental")
% (find-badioulowpage (+ 18 155) "II.3.2. Function of Appearing and Formal Definition")
% (find-badioulowtext (+ 18 155)      "2. Function of Appearing and Formal Definition")
% (find-badioulowpage (+ 18 156)         "The minimal requirement")
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% (find-badioulowpage (+ 18 157) "II.3.3. Equivalence-Structure and Order-Structure")
% (find-badioulowtext (+ 18 157)         "Equivalence-Structure and Order-Structure")

% (find-badioulowpage (+ 18 191) "Book III Greater Logic, 2. The Object")
% (find-badioulowpage (+ 18 193) "Introduction")
% (find-badioulowpage (+ 18 199) "Section 1 For a New Thinking of the Object")
% (find-badioulowpage (+ 18 199) "1 Transcendental indexing: the phenomenon")
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\newpage

Book II: Greater Logic

II.3. Algebra of the Transcendental

\subsection*{II.3.2. Function of Appearing and Formal Definition of the Transcendental}

(Page 157):

% (find-badioulowpage (+ 18 157) "The idea--a very simple one--")
% (find-badioulowtext (+ 18 157) "The idea--a very simple one--")

The idea --- a very simple one --- is that in every world, given two
beings $α$ and $β$ which are there, there exists a value $p$ of
$\Id(α,β)$. To say that $\Id(α,β)=p$ means that, with regard to their
appearing in that world, the beings $α$ and $β$ --- which remain
perfectly and univocally determined in their multiple composition ---
are identical to the $p$ degree', or are $p$-identical. The essential
requirement then is that the degrees $p$ are held in an
order-structure, so that for instance it can make sense to say that in
a fixed referential world, $α$ is more identical to $β$ than to $γ$.
In formal terms, if $\Id(α,β) = p$ and $\Id(α,γ) = q$, this means that
$p > q$.

\newpage

% (ph1p 5 "positional")
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% (favp 50 "functors-as-objects")
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The positional notations are explained in \cite[Section 1]{PH1} and
\cite[Section 7.12]{FavC}.

\newpage

% «2020aug04»  (to ".2020aug04")

Oi Caron! Tou tentando traduzir algumas definições da seção II.3.2.
Function of Appearing and Formal Definition of the Transcendental'' do
LoW pra uma terminologia mais padrão...

Eu deixei a câmera do celular aberta o tempo todo? Caramba...

Vou escrever umas duvidas aqui, aé quando você tiver tempo você ou me
responde ou me diz pra onde eu devo mandar...

Os degress of identity'' vão ser os elementos da álgebra de Heyting
dos valores de verdade do topos

Num dos exemplos que eu discuti com você e com o Gabriel a gente
começava com o house-shaped DAG'' $H$ que aparece aqui na pagina 27,

\url{http://angg.twu.net/LATEX/2017planar-has-1.pdf\#page=27}

E aí quando a gente montava o topos $\Set^H$ esse topos tinha 10
valores de verdade - a figura no topo da página 27.

Seja 1 o objeto terminal do topos $\Set^H$. Os valores de verdade
desse topos podem tanto ser vistos como os subobjetos desse 1 - lembra
que a gente pode usar a notacao $\Sub(A)$ pra falar do conjunto dos
subobjetos de um objeto $A$

quanto podem ser vistos como os morfismos do objeto 1 pro objeto
$\Omega$, onde $\Omega$ é o classificador.

Eu acho o $\Sub(1)$ mais fácil de visualizar.

Se a gente tem um objeto $A$ num topos os pontos de $A$ são os
morfismos do objeto 1 pro objeto $A$

Eu tou com a impressão de que quando o Badiou define $\Id(\alpha, \beta)$ esses $\alpha$ e $\beta$ (que na terminologia dele são
multiples'', se não me engano) são uma coisa um pouco mais

...porque tanto $\alpha$ quando $\beta$ podem ter um extent'' que é
um subobjeto do 1 que não é o próprio 1.

Na pagina 246 do PDF do LoW que eu tenho o Badiou define $𝐛E x := \Id(x,x)$

e um multiple'' $\alpha$ não é um morfismo de 1 para $A$, e sim um
morfismo de $𝐛E\alpha$ para $A$. Nao lembro a terminologia usual em
topos theory pra isso... acho que a gente chama de partial points''
ao inves de points''.

Se for isso eu posso fazer uns desenhos e mandar pro pessoal do seminário

Na verdade eu já tenho vários desses desenhos, é só reciclá-los...

\printbibliography

\end{document}

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