Reverse Mathematics (Summer 2026, TU Wien)

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Time: Tuesdays, 10am (academic time, i.e., really 10:15).
Place: TU Wien, Freihaus. Zeichensaal 1 (8th floor, green area, behind the elevators).

• March 10th. Introduction. The language of Second-Order Arithmetic. Second-Order Arithmetic. Arithmetical Comprehension. Recursive Comprehension.
   
• March 24th. The arithmetical and analytical hierarchies. The "Big Five" systems: RCA0, WKL0, ACA0, ATR0, Pi11-CA0. Proof that RCA0 does not imply WKL0. The Jockush-Soare Low Basis Theorem and proof that WKL0 does not imply ACA0.
   
• March 31st. No lecture (Easter).
   
• April 7th. No lecture (Easter).
   
• April 14th. Miscellaneous Q&A. Coding  in RCA0: first-order part (tuples, functions, etc.). Finite Sigma01-CA in RCA0. Proof that ACA0 is equivalent to "every function on the natural numbers has a range."
   
• April 23rd. The integers, rationals, and reals in RCA0. Proof that the reals are not countable in RCA0. Proof that the Bolzano-Weierstrass theorem is equivalent to ACA0.
   
• April 28th. No lecture on the occasion of the Gödel birthday colloquium. Location: Boecksaal, TU Wien.
  https://sites.google.com/view/goedel26/event
   
• April 30th. Compactness in RCA0. Proof that the Heine-Borel theorem is equivalent to WKL0. Complete, separable metric spaces. König's lemma for bounded-branching trees.
   
• May 5th. Full König's lemma. Finite-branching König's lemma. Gale-Stewart Games. WKL0 proves the determinacy of closed sets in Cantor space.
   
• May 19th. WKL0 is equivalent to the determinacy of closed sets in Cantor space. ACA0 is equivalent to the determinacy of finite games on integers. WKL0 is equivalent to the Gödel compactness and completeness theorem.
   
• May 26th. No lecture (Whit Tuesday).
   
• June 2nd. Transfinite induction in ACA0. ATR0. ATR0 proves the consistency of ACA0. Kleene's normal form for Pi11. Kleene-Brouwer orders. ACA0 proves the Pi11-completeness of wellfoundedness.
   
• June 9th. No lecture.
   
• June 16th.
   
• THURSDAY June 18th, 11:00–13:00. SEMINAR ROOM YELLOW, 7th floor. Make-up class.
   
• June 23rd.  

[1] S. Simpson. Subsystems of second-order arithmetic (1999). Cambridge University Press.

[2] D. Dzhafarov and C. Mummert. Reverse Mathematics (2022). Springer. 

[3] J. Stillwell. Reverse Mathematics: Proofs from the Inside Out (2018). Princeton University Press.

[1] C. G. Jockusch, Jr. and R. I. Soare. Π01 Classes and Degrees of Theories. Trans. Amer. Math. Soc. (1972).

[2] T. Nemoto, M. O. MedSalem, and K. Tanaka. Infinite games in the Cantor space and subsystems of second order arithmetic. Math. Log. Q. (1972).