Logical background of forcing

• Thesis title in Czech: Logické základy forcingu
• Thesis title in English: Logical background of forcing
• Key words: teorie množin, ZFC, forcing, interpretace, dokazatelnost, Peanova aritmetika, bi-interpretace, nestandardní model, spočetný tranzitivní model
• English key words: set theory, ZFC, forcing, interpretation, provability, Peano arithmetic, bi-interpretation, non-standard model, countable transitive model
• Academic year of topic announcement: 2010/2011
• Thesis type: diploma thesis
• Thesis language: angličtina
• Department: Department of Logic (21-KLOG)
• Supervisor: doc. Mgr. Radek Honzík, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 26.05.2011
• Date of assignment: 26.05.2011
• Administrator's approval: not processed yet
• Date and time of defence: 19.09.2013 12:00
• Date of electronic submission: 19.08.2013
• Date of proceeded defence: 19.09.2013
• Submitted/finalized: committed by student and finalized
• Opponents: RNDr. David Chodounský, Ph.D.

Guidelines

Práce zkoumá logické základy forcingu. Práce ukáže, do jaké míry lze forcing formalizovat tak, aby nemusel předpokládat, že existuje tranzitivní model ZFC. Dokonce lze forcingovou kontrukci vyjádřit i v PA (práce zkoumá meze tohoto vyjádření). Součástí práce je uvedení a důkaz základních forcingových vět, včetně několika příkladů (Cohen, Sacks). Součástí práce je přehled interpretačních technik (interpretace jedné teorie v druhé) a jejich souvislost s forcingem.

References

Kenneth Kunen, Set theory: An introduction to Independence proofs, Elsevier.
Tomas Jech, Set theory, Springer.
Vítězslav Švejdar, Logika: Neúplnost, složitost, nutnost, Academia.

Preliminary scope of work

The work studies the logical foundations of the method of forcing. The work shall show how to interpret forcing so that it does not need to presume the existence of a countable transitive model of ZFC. The forcing contrustion can be expressed even in PA (the work shall study the constraints of this expressibility in PA). The work shall include a review of the basic forcing theorems, and also examples of some of the best-known forcing constructions (eg. Cohen and Sacks forcing). The work shall review the technique of interpreting one theory in another, and will study the connections with forcing.

Preliminary scope of work in English

The work studies the logical foundations of the method of forcing. The work shall show how to interpret forcing so that it does not need to presume the existence of a countable transitive model of ZFC. The forcing contrustion can be expressed even in PA (the work shall study the constraints of this expressibility in PA). The work shall include a review of the basic forcing theorems, and also examples of some of the best-known forcing constructions (eg. Cohen and Sacks forcing). The work shall review the technique of interpreting one theory in another, and will study the connections with forcing.