jtotheizzoe:

How to Count Infinity

“Some infinities are bigger than other infinities.” - Hazel Grace Lancaster

Are there really more numbers between 0 and 1 than there are integers? MinutePhysics explores, in typically wonderful fashion.

(by minutephysics)

• Zeilberger, around 23 min, says there is a largest number, and the largest number+1 is 0
• “if the universe is finite, what’s outside of it”
• what if the only way to handle this was to recurse, to hide a, or multiple, universes within itself in order to maintain the logic of space and the paradox of “what is outside everything”

Problems in logic

• Hilbert’s Entscheidungsproblem.
• Type inference and type checking for the second-order lambda calculus (or equivalent).^[1]

[edit]Problems about abstract machines

• The halting problem (determining whether a Turing machine halts).
• Determining whether a Turing machine is a busy beaver champion (i.e., is the longest running among halting Turing machines with the same number of states).
• The mortality problem.
• Rice’s theorem states that for all nontrivial properties of partial functions, it is undecidable whether a machine computes a partial function with that property.

[edit]Problems about matrices

• The mortal matrix problem: determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. (This is known undecidable for a set of 7 or more 3 × 3 matrices, or a set of two 21 × 21 matrices.^[2])
• Determining whether a finite set of upper triangular 3 × 3 matrices with nonnegative integer entries generates a free semigroup.
• Determining whether two finitely generated subsemigroups of M_n(Z) have a common element.

[edit]Problems in combinatorial group theory

• The word problem for groups.
• The conjugacy problem.
• The group isomorphism problem.

[edit]Problems in topology

• Determining whether two finite simplicial complexes are homeomorphic.
• Determining whether a finite simplicial complex is (homeomorphic to) a manifold.
• Determining whether the fundamental group of a finite simplicial complex is trivial.

[edit]Problems in analysis

• For functions in certain classes, the problem of determining: whether two functions are equal; the zeroes of a function; whether the indefinite integral of a function is also in the class. For examples, see references in Stallworth and Roush, below. (These problems are not always undecidable. It depends on the class. For example, there is an effective decision procedure for the elementary integration of any function which belongs to a field of transcendental elementary functions, the Risch algorithm.)

• “The problem of deciding whether the definite contour multiple integral of an elementary meromorphic function is zero over an everywhere real analytic manifold on which it is analytic.” Its decidability would contradict the solution to Hilbert’s tenth problem.^[3]

[edit]Other problems

• The Post correspondence problem.
• The word problem in algebra and computer science.
• The word problem for certain formal languages.
• The problem of determining if a given set of Wang tiles can tile the plane.
• The problem whether a Tag system halts.
• The problem of determining the Kolmogorov complexity of a string.
• Hilbert’s tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers.
• Determining if a context-free grammar generates all possible strings, or if it is ambiguous.
• Given two context-free grammars, determining whether they generate the same set of strings, or whether one generates a subset of the strings generated by the other, or whether there is any string at all that both generate.
• Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions, or in a piecewise-linear flow in three dimensions.^{[citation needed]}
• Determining whether a λ-calculus formula has a normal form

Beautiful game logic.