Forcing Differentiable Functions

TL;DR

This paper explores the properties of differentiable functions in set theory, focusing on covering certain rectangles with smooth curves and analyzing differentiable isomorphisms between dense sets of real numbers.

Contribution

It introduces new results on the structure of differentiable functions within set theory, particularly regarding covering rectangles and isomorphisms between dense sets.

Findings

Covering Aleph_1 x Aleph_1 rectangles with smooth curves is studied.

Differentiable isomorphisms between Aleph_1-dense sets of reals are analyzed.

Results contribute to understanding the interplay between differentiability and set-theoretic structures.

Abstract

In various models of set theory, we consider covering Aleph_1 x Aleph_1 rectangles by countably many smooth curves, and we study differentiable isomorphisms between Aleph_1-dense sets of reals.