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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.
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?
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. And this led to the widespread claim that Gödel's Theorem implies the existence of mathematical statements that are true but unprovable—with their negations being false but unprovable. … 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.
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 .
Probably the most striking arise when one tries to apply traditional ideas of logic—and particularly notions of true and false. … In traditional logic there is always an operation of negation which takes any true statement, and makes it into a false one, and vice versa.
The idea is to set up an arithmetic statement that can be proved true if the evolution of a cellular automaton from a given initial condition makes a given cell be a given color at a given step, and can be proved false if it does not. … Such universality then implies Gödel's Theorem and shows that there must exist statements about arithmetic that cannot ever be proved true or false from its normal axioms.
As the picture below shows, there are 16 different possible operators that take two arguments and allow two values, say true and false. … Black stands for True ; white for False .
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.
The theorems are respectively: (1), (2) idempotence (laws of tautology) of And and Or, (3), (4) commutativity of And and Or, (5) law of double negation, (6), (7) absorption (redundancy) laws, (8) law of noncontradiction (definition of False), (9) law of excluded middle (definition of True), (10) de Morgan's law, (11), (12) associativity of And and Or, (13), (14) distributive laws.
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.