Summary of Godel’s Incompleteness Theorem

Kurt Godel presented a theory of mathematics that demonstrates that anything that can be measured, or contained, cannot be dependent upon itself for explanation or existence. The excerpts below from various sources give a summary of the implications of his theorem; for greater detail, the articles and book listed at the end are strongly recommended. (These statements were pulled from several sources; they are not my own material, other than the section on “Implications”)

“Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”

  1. Arithmetic is necessarily incomplete – because arithmetic axioms cannot be proven (axioms are assumptions about the nature of the subject matter; in arithmetic the subject is numbers and their behaviour)
  2. There are true statements of mathematics (theorems) which we can never formally know to be true. Associated facts: no theory can prove its own consistency
    ‘truth’ in a formal system/model cannot be defined within the system or model

Consequential fact: Any physical system subjected to measurement is capable of expressing elementary arithmetic. The Church-Turing thesis: a physical system that can express elementary arithmetic is not provable within the system, and is therefore incomplete. The universe is a physical system capable of expressing elementary arithmetic (it can be measured), and is therefore incomplete.


There are always more things that are true than that can be proven true.

All systems of logic or numbers must always rest on some unproven assumptions. No statement or system can prove itself true in isolation.

There is always something beyond anything that can be expressed arithmetically. The finite is bounded by the infinite.


Anything that can be quantified can be contained, and is therefore dependent upon an outside source for cause and definition.

The universe = all space + all time + all matter + energy = all of material reality.

The universe, as a physical system, can be expressed mathematically, is therefore incomplete, and must be explained / defined by something outside of itself, which is necessarily immaterial, or it would have to have been contained by the ‘system’ defined by all s+t+m+e; is necessarily an un-caused cause because any effect can be quantified and it is beyond that which is quantifiable – which also means it must be infinite because it is beyond quantification.


Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein