On the Uncountability Of

Abstract

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$ . We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from $[0,1]$ to ${\mathbb N}$ . Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$ . Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number.

On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

“…In this light, implicit in much of mathematical practise is the following most basic principle about countable sets: a set that can be mapped to N via an injection (or bijection) can be enumerated. This principle was studied in [73,86,88] as part of the study of the uncountability of R. In this paper, we continue the study of this principle in Reverse Mathematics (RM hereafter) and connect it to well-known 'household name' theorems due to Bolzano-Weierstrass, Cantor, Jordan, and Heine-Borel, as discussed in detail in Section 1.2. We assume basic familiarity with RM, also sketched in Section 1.3.1.…”

“…Since Cantor space with the lexicographic ordering and [0, 1] with its usual ordering are intimately connected, we take the former ordering to be fundamental. We have shown in [73] that BWC fun 0 is 'explosive' in that it yields the much stronger Π 1 2 -CA 0 when combined with the Suslin functional, i.e. higher-order Π 1 1 -CA 0 .…”

Big in Reverse Mathematics: the uncountability of the real numbers

“…Finally, the uncountability of R can be studied in numerous guises in higher-order RM. For instance, the following are from [57,58], where it is also shown that many extremely basic theorems imply these principles, while Z ω 2 cannot prove them. • For a countable set A ⊂ [0, 1], there is y ∈ [0, 1] \ A.…”

“…• NBI: there is no bijection from [0, 1] to N In particular, as shown in [58][59][60], the principles NBI and NIN are hard to prove in terms of conventional 2 comprehension, while the objects claimed to exist are hard to compute in terms of the other data, in the sense of Kleene's computability theory based on S1-S9 ( [38,47]). As shown in [67], this hardness remains if we restrict the mappings in NIN and NBI to well-known function classes, e.g.…”

“…based on bounded variation, Borel, upper semi-continuity, and quasi-continuity. Morover, many basic third-order theorems imply NIN or NBI, and the same at the computational level (see [57][58][59][60][65][66][67][68]). Finally, NIN and NBI seem to be the weakest natural thirdorder principles that boast all the aforementioned properties.…”