2 research outputs found
Fusible numbers and Peano Arithmetic
Inspired by a mathematical riddle involving fuses, we define the "fusible
numbers" as follows: is fusible, and whenever are fusible with
, the number is also fusible. We prove that the set of
fusible numbers, ordered by the usual order on , is well-ordered,
with order type . Furthermore, we prove that the density of the
fusible numbers along the real line grows at an incredibly fast rate: Letting
be the largest gap between consecutive fusible numbers in the interval
, we have for some constant
, where denotes the fast-growing hierarchy. Finally, we derive
some true statements that can be formulated but not proven in Peano Arithmetic,
of a different flavor than previously known such statements: PA cannot prove
the true statement "For every natural number there exists a smallest
fusible number larger than ." Also, consider the algorithm ": if
return , else return ." Then terminates on real inputs,
although PA cannot prove the statement " terminates on all natural inputs."Comment: Minor improvements. 26 pages, 5 figures, 3 table
Fusible numbers and Peano Arithmetic
Inspired by a mathematical riddle involving fuses, we define the "fusible
numbers" as follows: is fusible, and whenever are fusible with
, the number is also fusible. We prove that the set of
fusible numbers, ordered by the usual order on , is well-ordered,
with order type . Furthermore, we prove that the density of the
fusible numbers along the real line grows at an incredibly fast rate: Letting
be the largest gap between consecutive fusible numbers in the interval
, we have for some constant
, where denotes the fast-growing hierarchy. Finally, we derive
some true statements that can be formulated but not proven in Peano Arithmetic,
of a different flavor than previously known such statements: PA cannot prove
the true statement "For every natural number there exists a smallest
fusible number larger than ." Also, consider the algorithm ": if
return , else return ." Then terminates on real inputs,
although PA cannot prove the statement " terminates on all natural inputs.