— A Unified Perspective from Wittgenstein to Turing

Wittgenstein said in Tractatus Logico-Philosophicus:

The world is the totality of facts, not of things.

This sentence actually implies two completely different “storage containers”:

• Space container: Stores things (static structures, objects)
• Time container: Stores facts (dynamic events, evolutionary processes)

I. Completeness and Consistency

Using Gödel’s metamathematics:

• Time storage = Complete but inconsistent
• → Contains all facts that have happened, but those facts may contradict each other
• Space storage = Consistent but incomplete
• → Maintains logical coherence, but can only depict the finite set of things at a given moment
  

Conclusion: The capacity of the space container is far smaller than that of the time container.

Trying to use a static, formalized space structure to fully depict all facts in time is like using a finite model to hold infinite evolution — it is impossible.

II. The Root of Philosophy and Science Debates

In the history of human thought, many famous oppositions (reason vs. emotion, materialism vs. idealism, formalism vs. intuitionism, determinism vs. evolutionism) actually point to one core question:

Whether or not to include the time dimension in the axiomatic premise of analysis.

When you do not introduce time, the world is static, consistent, and formalizable;

Once you introduce time, the world becomes dynamic, uncertain, and difficult to be fully described by a closed system.

III. Turing’s Dual-System Modeling

In two key papers, Turing precisely modeled the two types of storage containers:

• On Computable Numbers, with an Application to the Entscheidungsproblem
• → Turing machine: Models space storage (formalized, deterministic, static deduction)
• Systems of Logic Based on Ordinals
• → Oracle machine + transfinite iteration: Models time storage (uncertain, dynamic, evolutionary sequences)
  

Turing’s breakthrough was that he first attempted to construct the world using both time and space containers:

• Space: Computable, verifiable structures
• Time: Dynamic evolution and unpredictable fact generation

IV. Conclusion and Insights

The world is a coupled system of time and space.

• Relying solely on the space container (formalism) cannot exhaust the facts of the world.
• Time storage must be combined with space storage to express the uncertainty of evolution while maintaining the verifiability of structure.
  

This also explains why more and more scientists and philosophers have realized:

You cannot use a static space model to encompass the entire dynamic world.

V. Time-Space Quadrant Diagram