Reverse mathematics has been the topic of my Honours project and my Masters thesis. Informally, reverse mathematics studies what axioms are required to prove certain theorems of “ordinary mathematics”. The Wikipedia page gives a nice summary; otherwise, for a more in-depth exposition, see my Honours report, or the books by Simpson or Stillwell.
My Honours project was under the supervision of Rod Downey and Noam Greenberg. We studied the reverse mathematical strength of Cousin’s lemma, a compactness principle arising in the study of the gauge integral.
Briefly, a gauge is a positive valued function \delta\colon [0,1] \to \mathbb{R}, and a tagged partition is a finite sequence
P = \langle 0 = x_0 < t_0 < x_1 < t_1 < \cdots < x_{\ell-1} < t_{\ell-1} < x_\ell = 1 \rangle
A partition P is \delta-fine if the open balls B \big( t_j, \delta(t_j) \big) cover [0,1] .
Cousin’s lemma. Every gauge \delta\colon [0,1] \to \mathbb{R} has a \delta-fine partition.
Reverse mathematics is traditionally done in second-order arithmetic, which is not powerful enough to talk about arbitrary functions f\colon \mathbb{R} \to \mathbb{R}. However, it can talk about any function which can be specified by countable information—examples include the continuous functions, Baire functions, and Borel functions. Rod, Noam and I proved the following results about Cousin’s lemma:
Theorem (Barrett, Downey, Greenberg). All implications are over \mathsf{RCA}_0.
- Cousin’s lemma for continuous functions is equivalent to \mathsf{WKL}_0.
- Cousin’s lemma for Baire 1 functions is provable in \Pi^1_1–\mathsf{CA}_0, and it implies \mathsf{ACA}_0.
- For n \geq 2, Cousin’s lemma for Baire n functions is provable in \Pi^1_1–\mathsf{CA}_0, and it implies \mathsf{ATR}_0.
These results are included in my Honours thesis, and will hopefully be published in the near future.
My Masters thesis is under the supervision of Dan Turetsky. We aim to study the reverse mathematics of certain topics in ring theory.