2022/03/10 by Sanders, Sam
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2203.05292
In his first set theory paper (1874), Cantor establishes the uncountability of ℝ. We study the latter in Kohlenbach's higher-order Reverse Mathematics, motivated by the observation that one cannot study concepts like `arbitrary mappings from ℝ to ℕ' in second-order Reverse Mathematics. Now, it was recently shown that the following statement: NIN: there is no injection from [0,1] to ℕ, is hard to prove in terms of conventional comprehension. In this paper, we show that NIN is robust by establishing equivalences between NIN and NIN restricted to mainstream function classes, like: bounded variation, semi-continuity, and Borel. Thus, the aforementioned hardness of NIN is not due to the quantification over arbitrary ℝ→ ℕ-functions in NIN. Finally, we also study NBI, the restriction of NIN to bijections, and the connection to Cousin's lemma and Jordan's decomposition theorem.