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Revision as of 00:04, 10 July 2008
, 00:04, 10 July 2008→Reality at the Threshold of Logic: HTML → TeX
|+ '''Table 5. A Bridge Over Troubled Waters'''
|+ '''Table 5. A Bridge Over Troubled Waters'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="center" | <math>\mbox{Linear Space}\!</math>
| align="center" | <math>\mbox{Liminal Space}\!</math>
| align="center" | <math>\mbox{Logical Space}\!</math>
|-
|-
|
|
<font face="lucida calligraphy">X</font><br>
<math>\begin{matrix}
{''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>}<br>
\mathcal{X}
& = & \{x_1, \ldots, x_n\} \\
\end{matrix}</math>
|
|
<font face="lucida calligraphy"><u>X</u></font><br>
<math>\begin{matrix}
{<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>}<br>
\underline\mathcal{X}
& = & \{\underline{x}_1, \ldots, \underline{x}_n\} \\
\end{matrix}</math>
|
|
<font face="lucida calligraphy">A</font><br>
<math>\begin{matrix}
{''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}<br>
\mathcal{A}
& = & \{a_1, \ldots, a_n\} \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
X_i
& = & \langle x_i \rangle \\
& \cong & \mathbb{K} \\
\end{matrix}</math>
|
|
<u>''X''</u><sub>''i''</sub><br>
<math>\begin{matrix}
{(<u>''x''</u><sub>''i''</sub>), <u>''x''</u><sub>''i''</sub>}<br>
\underline{X}_i
& = & \{(\underline{x}_i), \underline{x}_i \} \\
& \cong & \mathbb{B} \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
{(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br>
A_i
& = & \{(a_i), a_i \} \\
& \cong & \mathbb{B} \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
X \\
= & \langle \mathcal{X} \rangle \\
= & \langle x_1, \ldots, x_n \rangle \\
= & X_1 \times \ldots \times X_n \\
∏<sub>''i''</sub> ''X''<sub>''i''</sub><br>
= & \prod_{i=1}^n X_i \\
\cong & \mathbb{K}^n \\
\end{matrix}</math>
|
|
<u>''X''</u><br>
<math>\begin{matrix}
\underline{X} \\
= & \langle \underline\mathcal{X} \rangle \\
{‹<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>›}<br>
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\
= & \underline{X}_1 \times \ldots \times \underline{X}_n \\
∏<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br>
= & \prod_{i=1}^n \underline{X}_i \\
\cong & \mathbb{B}^n \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
A \\
= & \langle \mathcal{A} \rangle \\
= & \langle a_1, \ldots, a_n \rangle \\
= & A_1 \times \ldots \times A_n \\
∏<sub>''i''</sub> ''A''<sub>''i''</sub><br>
= & \prod_{i=1}^n A_i \\
\cong & \mathbb{B}^n \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
(hom : ''X'' → '''K''')<br>
X^*
& = & (\ell : X \to \mathbb{K}) \\
& \cong & \mathbb{K}^n \\
\end{matrix}</math>
|
|
<u>''X''</u>*<br>
<math>\begin{matrix}
(hom : <u>''X''</u> → '''B''')<br>
\underline{X}^*
& = & (\ell : \underline{X} \to \mathbb{B}) \\
& \cong & \mathbb{B}^n \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
(hom : ''A'' → '''B''')<br>
A^*
& = & (\ell : A \to \mathbb{B}) \\
& \cong & \mathbb{B}^n \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
(''X'' → '''K''')<br>
X^\uparrow
& = & (X \to \mathbb{K}) \\
('''K'''<sup>''n''</sup> → '''K''')
& \cong & (\mathbb{K}^n \to \mathbb{K}) \\
\end{matrix}</math>
|
|
<u>''X''</u>^<br>
<math>\begin{matrix}
(<u>''X''</u> → '''B''')<br>
\underline{X}^\uparrow
& = & (\underline{X} \to \mathbb{B}) \\
('''B'''<sup>''n''</sup> → '''B''')
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
(''A'' → '''B''')<br>
A^\uparrow
& = & (A \to \mathbb{B}) \\
('''B'''<sup>''n''</sup> → '''B''')
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
[<font face="lucida calligraphy">X</font>]<br>
X^\circ \\
[''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>]<br>
= & [\mathcal{X}] \\
(''X'', ''X''^)<br>
= & [x_1, \ldots, x_n] \\
(''X'' +→ '''K''')<br>
= & (X, X^\uparrow) \\
(''X'', (''X'' → '''K'''))<br>
= & (X\ +\!\to \mathbb{K}) \\
= & (X, (X \to \mathbb{K})) \\
('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> → '''K'''))<br>
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K})) \\
('''K'''<sup>''n''</sup> +→ '''K''')<br>
= & (\mathbb{K}^n\ +\!\to \mathbb{K}) \\
['''K'''<sup>''n''</sup>]
= & [\mathbb{K}^n] \\
\end{matrix}</math>
|
|
<u>''X''</u><sup>•</sup><br>
<math>\begin{matrix}
[<font face="lucida calligraphy"><u>X</u></font>]<br>
\underline{X}^\circ \\
[<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>]<br>
= & [\underline\mathcal{X}] \\
(<u>''X''</u>, <u>''X''</u>^)<br>
= & [\underline{x}_1, \ldots, \underline{x}_n] \\
(<u>''X''</u> +→ '''B''')<br>
= & (\underline{X}, \underline{X}^\uparrow) \\
(<u>''X''</u>, (<u>''X''</u> → '''B'''))<br>
= & (\underline{X}\ +\!\to \mathbb{B}) \\
= & (\underline{X}, (\underline{X} \to \mathbb{B})) \\
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
('''B'''<sup>''n''</sup> +→ '''B''')<br>
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
['''B'''<sup>''n''</sup>]
= & [\mathbb{B}^n] \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
[<font face="lucida calligraphy">A</font>]<br>
A^\circ \\
[''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]<br>
= & [\mathcal{A}] \\
(''A'', ''A''^)<br>
= & [a_1, \ldots, a_n] \\
(''A'' +→ '''B''')<br>
= & (A, A^\uparrow) \\
(''A'', (''A'' → '''B'''))<br>
= & (A\ +\!\to \mathbb{B}) \\
= & (A, (A \to \mathbb{B})) \\
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
('''B'''<sup>''n''</sup> +→ '''B''')<br>
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
['''B'''<sup>''n''</sup>]
= & [\mathbb{B}^n] \\
\end{matrix}</math>
|}<br>
|}<br>
