Derived sequences and reverse mathematics

✍ Scribed by Jeffry L. Hirst

• Publisher: John Wiley and Sons
• Year: 1993
• Tongue: English
• Weight: 412 KB
• Volume: 39
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19930390148

✦ Synopsis

Abstract

One of the earliest applications of transfinite numbers is in the construction of derived sequences by Cantor [2]. In [6], the existence of derived sequences for countable closed sets is proved in ATR~0~. This existence theorem is an intermediate step in a proof that a statement concerning topological comparability is equivalent to ATR~0~. In actuality, the full strength of ATR~0~ is used in proving the existence theorem. To show this, we will derive a statement known to be equivalent to ATR~0~, using only RCA~0~ and the assertion that every countable closed set has a derived sequence. We will use three of the subsystems of second order arithmetic defined by H. Friedman ([3], [4]), which can be roughly characterized by the strength of their set comprehension axioms. RCA~0~ includes comprehension for Δ definable sets, ACA~0~ includes comprehension for arithmetical sets, and ATR~0~ appends to ACA~0~ a comprehension scheme for sets defined by transfinite recursion on arithmetical formulas. MSC: 03F35, 54B99.