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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 13:14, 21 May 2008
, 13:14, 21 May 2008→Formal development: HTML --> TeX
'''The Extended Universe of Discourse'''
'''The Extended Universe of Discourse'''
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' E<font face="lucida calligraphy">A</font> as:
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as:
: E<font face="lucida calligraphy">A</font> = <font face="lucida calligraphy">A</font> ∪ d<font face="lucida calligraphy">A</font> = {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>, d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
: <math>\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A} = \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math>
This supplies enough material to construct the ''differential extension'' E''A'', or the ''tangent bundle'' over the initial space ''A'', in the following fashion:
This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A</math>, or the ''tangent bundle'' over the initial space <math>A\!</math>, in the following fashion:
:{| cellpadding=2
:{| cellpadding=2
| E''A''
| <math>\operatorname{E}A</math>
| =
| =
| ''A'' × d''A''
| <math>A \times \operatorname{d}A</math>
|-
|-
|
|
| =
| =
| 〈E<font face="lucida calligraphy">A</font>〉
| <math>\langle \operatorname{E}\mathcal{A} \rangle</math>
|-
|-
|
|
| =
| =
| 〈<font face="lucida calligraphy">A</font> ∪ d<font face="lucida calligraphy">A</font>〉
| <math>\langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle</math>
|-
|-
|
|
| =
| =
| 〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>, d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>〉,
| <math>\langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math>
|}
|}
thus giving E''A'' the type '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>.
thus giving <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>
Finally, the tangent universe E''A''<sup> •</sup> = [E<font face="lucida calligraphy">A</font>] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features E<font face="lucida calligraphy">A</font>:
Finally, the tangent universe E''A''<sup> •</sup> = [E<font face="lucida calligraphy">A</font>] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features E<font face="lucida calligraphy">A</font>: