Homework for 25.3.2020 and 1.4.2020 (virtual teaching)
You should write a one-page of summary of different positions on different criteria for accepting axioms of mathematics, in particular of set theory (should they be evident? should usefulness count? should we decide based on experience or intrinsic value?). Use Maddy's paper Believing the Axioms I. Additionally, you should write a one-page summary on arguments for and against CH. Focus on arguments which are close to your focus (mathematical, or philosophical).
You should continue reading Maddy's paper Believing the Axioms I, focus on the Continuum Hypothesis (CH). What are the most comment arguments for or against accepting CH? Think about equivalent formulations of CH.
(old) We will discuss this on a Zoom meeting on 25.3. 11:30, link https://zoom.us/j/3062368666
We will continue discussing this on a Zoom meeting on 1.4. 11:30, link https://zoom.us/j/3062368666 (please have your qeustions ready).
Consider the following points (Maddy's first paper):
- Recall that CH is equal to the fact that for all infinite subsets X of R, either X is at most countable, or it has the same size as R. We say that CH holds for a set A of subsets of R if for every infinite X from A, either X is at most countable or it has the same size as R.
- Cantor's views, secion II.1: In favour of CH due to theorems which claim that CH holds for "simple" sets of reals (open, closed, Borel,...). Cantor thought that a proof could be found which would show that for "all" sets of reals.
- Cohen, section II.2. Development of forcing (see my survey paper Basic Set theory, (semi)popular explanation below "A quick guide to independence results" for a gentle introduction to forcing). CH independent over ZFC. However, ZFC + V=L implies CH. So perhaps V = L is a good candidate for deciding many statements? (forcing does not work for ZFC + V = L).
- Section II.3.1 Partial results (in favour): CH holds for all Borel (even analytic sets). Definitions of these concepts can be found in all set theory textbooks (Balcar+Stepanek, Jech, Kanamovi).
- Section II.3.2. Effectiveness.
- Section II.3.3. Godel against (the reasons are technical, we can discuss them at some point if you are interested; for instance "absolute zero" should be understood as "strong measure zero sets" - uncountable strong measure zero sets are considered "counterintuitive", but their existence is implied by CH; it is consistent that CH fails and there are no uncountable strong measure zero sets).
- Section II.3.4. Restrictive (against)
- Section II.3.5-12. for or against...
Bottom line: which of these arguments have a chance to be universally accepted? Is there a connection of CH with large cardinals or other "universally accpatable axioms"?
For the class on April 8, 11:30, please read the definitions and the context for the following large cardinals: inaccessible, Mahlo, weakly compact, measurable. You may use the paper "Large cardinals and CH" or Maddy's Believing Axioms 1, which can be downloaded below. Think whether the existence of such cardinals seems to you "intuitively true", i.e. whether you think they are good cadidates for new axioms of ZFC. I will be happy to read your ideas about this, but this is not a mandatory homework this time.
We will continue discussing this on a Zoom meeting on 8.4. 11:30, link https://zoom.us/j/3062368666 (please have your qeustions ready).
Please skim the papers below, optionally also read up on the epsilon calculus in the link provided. We'll meet again on April 15, 11:30, with the usual link.
Based on your preferences, please browse through the papers and a book listed below.
As we discussed, you should write a paper roughly 10 pages long about a topic of your choosing, possibly based on the papers we discussed in this class. You should try to make it a well-researched paper which could (after some improvements) be submitted to a journal.
Deadline for submission: one week before an exam. But I recommend to submit by the end of the semester when some of the ideas are still "fresh".
There will be no regular class on May 20. However, we can meet individually to discuss your final paper. We can meet either over Zoom in person, if we find a convenient time. Write me an email to arrange a date and time for consultation.