Infinitary logic (Winter 2025, TU Wien)

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Time: Fridays 1pm–3pm (new time).
Place: Besprechungsraum on the 5th floor in the green area of the Freihaus building of TU Wien.

Exceptions: on Friday, November 14th the lecture will take place in seminar room green 06B.

• October 10th, 2025. Introduction. Recap (first-order logic: compactness, completeness, the Löwenheim-Skolem theorem; ultrafilters; ordinals and cardinals). Abstract definition of a logic. Some examples of logics. The downward Löwenheim-Skolem theorem for L_{\omega_1,\omega}.
   
• October 17th, 2025. More examples of logics. Consistency properties. The Model Existence Theorem for L_{\omega_1,\omega} [4].
   
• October 24th, 2025. Proof of the The Model Existence Theorem. Hanf numbers.
   
• October 31st, 2025. Lower bound for the Hanf number of L_{\omega_1,\omega}. Ramsey's Theorem. The Erdős-Rado Theorem [2].
   
• November 14th, 2025. The Hanf number of L_{\omega_1,\omega} [4].
   
• November 21st, 2025. Measurable cardinals [2].
   
• November 28th, 2025. Measurable cardinals (continued). The Hanf-Tarski number of L_{\omega_1,\omega}.
   
• December 5th, 2025. The Hanf-Tarski number of L_{\omega_1,\omega} (continued, [5]). Compactness numbers. Strongly compact cardinals. Strongly compact cardinals and the compactness of L_{\kappa,\kappa}, [2].
   
• December 12th, 2025. \omega_1-strongly compact cardinals. The Bagaria-Magidor theorem: the compactness number of L_{\omega_1,\omega} [7].
   
• January 9th, 2025. Back-and-forth families. Karp's theorem. Ehrenfeucht-Fraïssé games. [4]
   
• January 16th, 2025. Scott families. Scott's theorem. \omega-logic: soundness and completeness. [4]
   
• January 23rd, 2025. Epilog: Barwise completeness. Barwise compactness. The Barwise extension theorem. 

[1] J. L. Bell. Infinitary Logic. In: Stanford Encyclopedia of Philosophy (2023). Available at https://plato.stanford.edu/entries/logic-infinitary/

[2] T. Jech. Set Theory: The Third Millenium Edition (2023). Springer.

[3] C. C. Chang and H. J. Keisler. Model Theory (1990). North-Holland.

[4] D. Marker. Lectures on Infinitary Model Theory (2013). Cambridge University Press.

[5] J. P. Aguilera. Local Hanf-Tarski numbers. In: Trans. Amer. Math. Soc., in press.

[6] J. Bagaria and M. Magidor. Group Radicals and Strongly Compact Cardinals. In: Trans. Amer. Math. Soc. (2013).

[7]. V. Gitman and J. Osinski. Upward Löwenheim-Skolem-Tarski numbers for abstract logics. In: Ann. Pure Appl. Logic (2025).