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Much of this section will be dumped at <website front-end for a mathematical formal proof system>{full} instead.


Proof assistant

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A proof in some system for the <formalization of mathematics>.


Formal proof

{disambiguate=mathematics}

A <theorem> that is not very important on its own, often an intermediate step to proving something that the author feels deserves the name "<theorem>".


Lemma

Lemma (mathematics)

{disambiguate=mathematics}

= Set
{synonym}

Intuitively: unordered container where all the values are unique, just like C++ `std::set`.

More precisely for set theory <formalization of mathematics>:
* everything is a set, including the elements of sets
* string manipulation wise:
  * `{}` is an empty set. The natural number `0` is defined as `{}` as well.
  * `{{}}` is a set that contains an empty set
  * `{{}, {{}}}` is a set that contains two sets: `{}` and `{{}}`
  * `{{}, {}}` is not well formed, because it contains `{}` twice


Set (mathematics)

{disambiguate=mathematics}
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= Function
{synonym}

<Set (mathematics)> of <ordered pairs>. That's it! This is illustrated at: https://math.stackexchange.com/questions/1480651/is-fx-x-1-x-2-a-function/1481099#1481099


Function

Function (mathematics)

Function space

Number

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= Complex
{synonym}

An <ordered pair> of two <real numbers> with the complex addition and multiplication defined.

Forms both a:
* <division algebra> if thought of <\R^2> with complex multiplication as the bilinear map of the <algebra over a field>[algebra]
* <field (mathematics)>


Complex number

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<Sets> are unordered, but we can use them to create ordered objects, which are of fundamental importance. Notably, they are used in the definition of <functions>.


Ciro Santilli @cirosantilli 34

 Children: Formalization of mathematics