MATH 141A: Mathematical Logic I

Jump to Today

Please fill out the evaluations. Your feedback is valuable, helpful, and important!
I will appreciate it if everyone can submit the Q form. Thanks!

Notes - Updated 4/26.

Please feel free to email me if you have any questions, or to set up a meeting. - Assaf

Zoom link - Please ask if you need Zoom accomodations.

Contact:
- Assaf - shani@math.harvard.edu
- Joanna - jboyland@college.harvard.edu

A review on Equivalence Relations

References:
- Notes by Professor W. Hugh Woodin and Professor Theodore A. Slaman.
- Herbert B. Enderton - A Mathematical Introduction to Logic. (Online access.)
- David Marker - Model Theory: An Introduction. (Online access.)
- Wilfrid Hodges - Model Theory. (Online access)

Further further references: I will add here some advanced (research level) topics. These are not things that can be "easily read". Feel free to talk to me if you are interested in any.

  • Model theory and Machine learning.
    • There is an interesting interaction between model theoretic questions and questions about learnability. (Both are deeply tied with questions in finite combinatorics.)
    • Here is one recent paper on the subject. Here is another. I also found these slides on the topic.
  • Some deep connections between model theory and set theory.
    • A natural continuation of the topics we covered in class can be found in graduate model theory books, such as Marker's book and Hodges' book.  These direction often require some background in set theory as well.
    • Here is a more recent survey article about recent research directions in model theory which involve deeply set theoretic questions.
      [This would be relevant if you are interested in abstract model theory and set theory.]
  • Ramsey theory + Fraisse theory = topological dynamics.
    • We briefly mentioned that structures like the rational order and the random graph are particular examples of a general phenomenon of "Fraisse limits" of finite objects. (See notes for reading references.)
      It turns out that the dynamics of the group of automorphisms of such structure rely on generalizations of Ramsey's theorem. For example, Ramsey's theorem that we proved yields some restrictions on the group of automorphisms of LaTeX: (\mathbb{Q},<).
    • This was introduced about 20 years ago in this paper. You can find here a more recent survey (though very advanced).
      [To read these you would need a strong background in topology and analysis.]
  • Algebra and model theory.
    • Model theory has extremely tight connections with, and many applications to, Algebra, and field theory in particular. One place to start is Chapter 3 of Marker's book.
      [This would require some familiarity with algebra, but only the basics of fields, rings, and groups, would suffice to see some interesting applications of model theory.]
  •  Continuous Logic or "Model theory for metric structures".
    • Much of discrete mathematics (groups, graphs, fields,...) can be naturally formulated in the model theory we are discussing (model theory of "first order logic"). What about non-discrete objects, such as complete metric spaces, Hilbert spaces, Banach spaces, measure spaces? It turns out that by changing the "first order logic" (but not too drastically) to "continuous logic", these structures can be naturally presented, and studied using model theoretic techniques.
    • It was introduced in this paper. Roughly speaking, the first few chapters can be seen as parallel to our course, just for this "continuous logic" replacing our "first order logic". You can also find here a high-level advanced mini-course (focusing on rather more advanced topics).

Some non-mathematics writings (written by logicians):

 

Resubmission policy: See the syllabus.
- Recall that for the resubmission you are not allowed to collaborate or use any sources (other than our notes).
- Resubmission is per question. You do not need to resubmit everything.
- Resubmissions are to be handed directly to me (Assaf).
- Your resubmission must be clear, correct, and precise. (This is true for the first submission too, but even more so for the resubmission.)

Class meetings: Tuesday and Thursday 1:30pm - 2:45pm. 
Place: Science Center 110. 

Syllabus 

 

Course Summary:

Date Details Due