# 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 indepen
dent to the set at each time, we get a chain {v1} <= {v1,v2} <= {v
1,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 a
n axiom.
This kind of thinking is somewhat universal. Why we use an axiomat
ic system is an example. To prove a proposition, we must have assu
mptions. Then we have to prove those assumptions. This forms a cha
in.
... -> P2 -> P1 -> P0 (P0 is the original proposition)
Will the chain end? I don't know. If you think it won't end, you w
ill get an infinite regression. If you think it'll end, you will g
et 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 oth
er thing. That is "Monad". Pythagoreans believed that it is the di
vinity 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 th
> e Word was God.
>
> KJV, John 1:1
Word is a translation of Logos. Thomas Aquinas said the Logos is l
iterally god. Corpus Hermeticum says it is an idea of god. Pythago
reans, Gnostics, Neoplatonists also believed similar things.
The logos, axioms, Zorn's lemma are all just variations of Indo-Eu
ropean tradition. You can deal with them seperately, but knowing t
he bigger picture is always astonishing.