응시, 원경의 지평

<Contemplative Contemplation - The Horizon in a distant view>

ENDLESS OPENNESS

According to the Gödel's Incompleteness Theorem

If F is a consistent formal system that is capable of expressing arithmetic,

then there exists a statement G such that:

This tells us that there are true statements G

that cannot be proven within the system F.

Let 𝕂 represent the body of knowledge within a formal system F.

Gödel's theorem implies:

This means there exists a true statement G

that is not provable within the system F.

ITERATIVE EXPANSION

As we expand our system F to a new system F' to include G, there will always be new statements G' that remain unprovable:

Each expansion leads to a new system where new true but unprovable statements exist, symbolizing the endless openness.

HIERARCHICAL SYSTEMS

The idea of an infinite hierarchy of systems emphasizes the ever-growing horizon of knowledge.

LIMIT OF KNOWLEDGE

As n approaches infinity, the union of all these systems still leaves some truths unprovable:

The conclusion that even the union of all these systems might not encompass complete knowledge reinforces the concept of "endless openness.”