Ciro Santilli @cirosantilli 37

Basically ensuring that good AI alignment allows us to survive the singularity.

A theorem is said to be independent from a set of axioms if it cannot be proven neither true nor false from those axioms.

It or its negation could therefore be arbitrarily added to the set of axioms.

Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.

A unique projective space can be defined for any vector space.

The projective space associated with a given vector space $V$ is denoted $\mathbf{P}(V)$.

The definition is to take the vector space, remove the zero element, and identify all elements that lie on the same line, i.e. $\vec{v}=\lambda \vec{w}$

The most important initial example to study is the real projective plane.

Elements of a Lie algebra can (should!) be seen a continuous analogue to the generating set of a group in finite groups.

For continuous groups however, we can't have a finite generating set in the strict sense, as a finite set won't ever cover every possible point.

But the generator of a Lie algebra can be finite.

And just like in finite groups, where you can specify the full group by specifying only the relationships between generating elements, in the Lie algebra you can almost specify the full group by specifying the relationships between the elements of a generator of the Lie algebra.

This "specification of a relation" is done by defining the Lie bracket.

The reason why the algebra works out well for continuous stuff is that by definition an algebra over a field is a vector space with some extra structure, and we know very well how to make infinitesimal elements in a vector space: just multiply its vectors by a constant $c$ that cana be arbitrarily small.

TODO concrete example, please...

A set $S$ plus any number of functions $f_i:S\times S\to S$, such that each $f_i$ satisfies some properties of choice.

Key examples:

A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

Metric created by IonQ.

There are a few related concepts that are called infinity in mathematics: