In a multiway system, one can imagine identifying "true" with a string consisting of a single black element. … For the picture implies that both and its negation can be proved to be true statements. … And so what Multiway systems starting from a single black element that represents True.

And what this means is that the set of statements that can be proved true will never overlap with the set that can be proved false. But can every possible statement that one might expect to be true or false actually in the end be proved either true or false? The network of statements that can be proved true using the axiom system for logic from page 775 . p ⊼ (p ⊼ p) is the simplest representation for True when logic is set up using the Nand operator ⊼ .

For as soon as there are just two objects that both satisfy the constraints but for which there is some statement that is true about one but false about the other it immediately follows that at least this statement cannot consistently be proved true or false, and that therefore the axiom system must be incomplete. … For given any statement that cannot be proved from the axioms there must be distinct objects for which it is true, and for which it is false. … But if an axiom system is close to complete—so that the vast majority of statements can be proved true or false—then it is almost inevitable that the different kinds of objects that satisfy its constraints must differ only in obscure ways.

Yet often it seemed inevitable just from the syntactic structure of statements (say as well-formed formulas) that each of them must at some level be either true or false. … In some cases statements can in effect have default truth values—so that showing that they are unprovable immediately implies, say, that they must be true. … And thus for example the Continuum Hypothesis in set theory is unprovable but could be either of true or false: it is just independent of the axioms of set theory.

general features that somehow capture the essence of true intelligence, independent of the particular details of human intelligence. … At first, all of these might seem like reasonable indicators of true intelligence. … So given all of this is there any way to define a general notion of true intelligence?

Yet it is rather close to being complete—since as we saw earlier one has to go through at least millions of statements before finding ones that it cannot prove true or false. … There is again incompleteness, but now there is much more of it, for even statements as simple as x+y  y+x and x+0  x cannot be proved true or false from the axioms. … But in a field like set theory this is less true.

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. … A consequence of this is that there can be no finite procedure that always decides whether a given statement is true—making the system what is known as essentially undecidable.

If the last 2 axioms are dropped any statement can readily be proved true or false essentially just by running rule 110 for a finite number of steps equal to the number of nested ↓ plus 〈 … 〉 in the statement. … One can establish that the statement at the bottom on the right cannot be proved either true or false from the axioms by showing that it is true for some initial conditions and false for others. … So this means that the statement is false if the initial condition is and true if the initial condition is .

Truth and falsity [in formal systems] The notion that statements can always be classified as either true or false has been a common idealization in logic since antiquity. … An example is x + y  z , which cannot reasonably be considered either true or false unless one knows what x , y and z are. … In Mathematica functions like TrueQ and IntegerQ are set up always to yield True or False —but just by looking at the explicit structure of a symbolic expression.

All the statements in the top block above can be proved true from the axiom system. The bottom-right statement, however, cannot be proved either true or false.