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  1. Can someone explain in layman's terms or at least give a summary of the key differences of approaches between Woodin and Sy Friedman regarding set-theoretic truths?

  2. After this great debate, are there any progress being made?

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It is only a half-joke to say that for a layman, there is no difference between their approaches. Take the following question for example: Does there exist an uncountable subset of $\mathbb R$ that cannot be put into bijection with $\mathbb R$? Both Friedman and Woodin would say that this is a question that might admit a definite yes-or-no answer, but that the jury is still out and we need to do a whole lot more technical set-theoretic work before we will be in a position to say whether it does admit a definite answer, and if so, whether the answer is yes or no. This already sets them aside from many others, e.g.,

  1. formalists who say that the question does not admit a definite yes-or-no answer, or

  2. platonists who say that the question has a definite answer but who also say that we already know that we will never know the answer.

To even state the differences between their approaches requires considerable technical machinery. Friedman's approach is called the "Hyperuniverse Program." Roughly speaking, his "hyperuniverse" is the space of all countable transitive models of ZFC, and the idea is that these universes are not all on an equal footing; e.g., some models are in a sense "maximal" and this clues us to which set-theoretic statements are true. Woodin on the other hand may be thought of as taking the inner model program as a starting point and arguing that the mathematical evidence points towards something called $\Omega$-logic as a natural foundation for set-theoretic truth. I do not think that too many further details can really be accurately described in "layman's terms" because the technical prerequisites are considerable.

As for whether there has been "progress," both approaches lead to a plethora of technical mathematical questions, and although I do not follow this area, I expect that progress has been made in the same way that any subfield of mathematics makes progress. But if that's what you mean by progress then again I am not sure that it is possible to summarize it "for a layman." On the other hand, a layman could perhaps follow some of the philosophical debates about truth, but then it is less clear (at least for most mathematicians) what "progress" means in the philosophical realm.

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    $\begingroup$ Maybe it's worth mentioning that Woodin changed his mind not too long ago on what the answer is to the question "Does there exist an uncountable subset of R that cannot be put into bijection with R?" This may or may not be seen as "progress". $\endgroup$ – David Roberts Sep 23 '17 at 6:17
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    $\begingroup$ Uh, I’m a Platonist, and I’m hardly sure of the second part of (2). I believe Gödel, in “What is Cantor’s Continuum Problem,” expressed hope that it might turn out to be false, but my copy is thousands of miles away. So which Platonists do believe 2.2? $\endgroup$ – Flash Sheridan Sep 23 '17 at 13:11
  • $\begingroup$ Following up on what David Roberts said, my understanding was that Sargsyan's thesis had damaged Woodin's $\Omega$-logic idea (although I don't know the details). $\endgroup$ – Noah Schweber Sep 23 '17 at 14:03
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    $\begingroup$ @FlashSheridan : My list was not meant to be comprehensive and I did not mean to imply that all platonists fall into category 2. On the other hand, back in Goedel's day, it seemed that large cardinals might suffice to settle CH, but the Levy-Solovay theorem about the behavior of large cardinals under forcing makes that seem unlikely. I think that many Platonists who are only prepared to accept new set-theoretic axioms that seem "obviously true" are pessimistic about finding any such axioms that would settle CH. $\endgroup$ – Timothy Chow Sep 23 '17 at 19:28

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