But normally such a statement cannot be proved true or false within the system itself.

The shells are the following types: first row: Elliot's volute, vexillate volute, lettered cone; second row: music volute, banded marble cone, tent olive; third row: bough cone, textile cone, false melon volute ( Livonia mammilla ).

With black and white interpreted as True and False , the forms of operators shown here correspond respectively to And , Equal , Implies and Nand .

Essential incompleteness [in axiom systems] If a consistent axiom system is complete this means that any statement in the system can be proved true or false using its axioms, and the question of whether a statement is true can always be decided by a finite procedure. If an axiom system is incomplete then this means that there are statements that cannot be proved true or false using its axioms—and which must therefore be considered independent of those axioms.

Implementation [of conserved quantity test] Whether a k -color cellular automaton with range r conserves total cell value can be determined from Catch[Do[ (If[Apply[Plus, CAStep[rule, #] - #] ≠ 0, Throw[False]] &)[ IntegerDigits[i, k, m]], {m, w}, {i, 0, k m - 1}]; True] where w can be taken to be k 2r , and perhaps smaller.

Complements of recursively enumerable sets are characteristically associated with Π 1 statements of the form ∀ t ϕ [t] —an example being whether a given system never halts. ( Π 1 and Σ 1 statements are such that if they can be shown to be undecidable, then respectively they must be true or false, as discussed on page 1167 .) … (Showing that a statement with n ≥ 1 is undecidable does not establish that it is always true or always false.)

And applying Complement[s, Intersection[a, b]] to these two elements gives the same results and same equivalences as a ⊼ b applied to True and False . … But all this actually does is to force there to be only two objects analogous to True and False .)

Statements in Peano arithmetic Examples include: • √ 2 is irrational: ¬ ∃ a ( ∃ b (b ≠ 0 ∧ a × a  ( Δ Δ 0) × (b × b))) • There are infinitely many primes of the form n 2 + 1 : ¬ ∃ n ( ∀ c ( ∃ a ( ∃ b (n + c) × (n + c) + Δ 0  ( Δ Δ a) × ( Δ Δ b)))) • Every even number (greater than 2) is the sum of two primes (Goldbach's Conjecture; see page 135 ): ∀ a ( ∃ b ( ∃ c (( Δ Δ 0) × ( Δ Δ a)  b + c ∧ ∀ d ( ∀ e ( ∀ f ((f  ( Δ Δ d) × ( Δ Δ e) ∨ f  Δ 0) ⇒ (f ≠ b ∧ f ≠ c))))))) The last two statements have never been proved true or false, and remain unsolved problems of number theory.

But since it is independent of the axioms of arithmetic there must be objects that still satisfy the axioms but for which it is false.

The bottom-right statement, however, cannot be proved either true or false.