# Zorn's Lemma, Axioms, Logos

In mathematics, a lemma called "Zorn's lemma" exists. It basically
says: Every chain has its maximal element. For example, in linear
algebra, [basis] = [linearly independent] + [generating]. Hence if
we start with a vector, and add a vector which is linearly independent
to the set at each time, we get a chain {v1} <= {v1,v2} <= {v1,v2,v3}
<= ... Zorn's lemma guarantees this chain ends, and that end is a
basis. In fact, this is equivalent to Zorn's lemma. Zorn's lemma
is also equivalent to Axiom of Choice, so it is actually an axiom.

This kind of thinking is somewhat universal. Why we use an axiomatic
system is an example. To prove a proposition, we must have assumptions.
Then we have to prove those assumptions. This forms a chain.

... -> P2 -> P1 -> P0

P0 is the original proposition. Will the chain end? I don't know.
If you think it won't end, you will get an infinite regression. If
you think it'll end, you will get an axiom. If you think chains
are interdependent, you will get a circular logic, like Buddhists
believe.

However, Indo-European tradition says a chain will end. The end is
an axiom. So math is the self-expansion of the axiomatic system.

Indo-European tradition also says that one axiom implies every
other thing. That is "Monad". Pythagoreans believed that it is the
divinity itself. The monad is in the logos. The two are similar,
but the monad is like a spark, while the logos is like a fire. A
very famous verse in the bible about the logos exists.

> In the beginning was the Word, and the Word was with God, and
> the Word was God.
>
> KJV, John 1:1

Word is a translation of Logos. Thomas Aquinas said the Logos is
literally god. Corpus Hermeticum says it is an idea of god.
Pythagoreans, Gnostics, Neoplatonists also believed similar things.


The logos, axioms, Zorn's lemma are all just variations of
Indo-European tradition. You can deal with them seperately, but
knowing the bigger picture is always astonishing.