I suspect that the same will be true of the basic rule for the universe.

Perhaps more surprisingly, the same is also true for ordinary tag systems.

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 .)

But while this may sometimes be true—perhaps as a consequence of the Central Limit Theorem—it is rarely checked, making it likely that many detailed inferences are wrong.

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.

This will happen whenever FractionalPart[Log[b, a[n]]] is uniformly distributed, which, as discussed on page 903 , is known to be true for sequences such as r n (with Log[b, r] irrational), n n , n!

Boole identified 1 with True and 0 with False , then noted that theorems in logic could be stated as equations in which Or is roughly Plus and And is Times —and that such equations can be manipulated by algebraic means. … The tradition in philosophy and mathematical logic has more been to take axioms to be true statements from which others can be deduced by the modus ponens inference rule {x, x  y}  y (see page 1155 ). … The question of whether any particular statement in basic logic is true or false is always formally decidable, although in general it is NP-complete (see page 768 ).

And while it is indeed true that for almost every rule the specific pattern produced is at least somewhat different, when one looks at all the rules together, one sees something quite remarkable: that even though each pattern is different in detail, the number of fundamentally different types of patterns is very limited.

But what must be true is that there can never be any ambiguity about what replacement will eventually be made in any given part of the system.

But must it in the end actually be true that the underlying rules for our universe force there to be a unique perceived history?