Cancellation and Schröder–Bernstein for Torsion Abelian Groups

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Work in second-order arithmetic over $\mathsf{RCA}_0$, the base theory with $\Delta^0_1$ comprehension and $\Sigma^0_1$ induction. The strength of a principle means which subsystems prove it and which subsystems it implies over that base; equivalence requires both implications. The theory $\Pi^1_1\text{-}\mathsf{CA}_0$ adds comprehension for formulas with a universal block of set quantifiers and an arithmetical matrix. The theory $\mathsf{ATR}_0$ permits arithmetical transfinite recursion along every coded countable well-order. A countable abelian group is torsion if every element has finite order. A direct summand $G$ of $H$ means $H\cong G\oplus K$ for some group $K$. A group is divisible if for every positive integer $n$ and element $a$ there is $b$ with $nb=a$; it is reduced if it has no nonzero divisible subgroup.

Statement recorded in the supplied source.

Is each of the following principles equivalent to $\Pi^1_1\text{-}\mathsf{CA}_0$? (i) For all countable torsion abelian $G,H$, $G\oplus G\cong H\oplus H$ implies $G\cong H$. (ii) For all such $G,H$, if each is a direct summand of the other, then $G\cong H$. In the source’s accompanying conjecture, is each principle restricted to reduced torsion abelian groups equivalent to $\mathsf{ATR}_0$? [S2]

Short English statement

Do the two cancellation principles for countable torsion abelian groups each require analytic comprehension? When the groups have no divisible part, do they instead each require arithmetic transfinite recursion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7500112 (AMR-074-0112). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Antonio Montalbán, Open Questions in Reverse Mathematics, author preprint dated 17 August 2010, Question 12, printed p. 6. https://math.berkeley.edu/~antonio/papers/questionsRM.pdf.