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Joel David Hamkins
Mathematics and philosophy of the infinite
Mathematics and philosophy of the infinite

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Material set theory

One sometimes sees the term "material set theory" used to refer to the subject otherwise usually known as set theory.

I wanted to mention, especially for those set theorists not aware of the connotations the term carries, that this is not really a neutral term. Rather, this term was introduced by category theorists to convey a certain part of their philosophical criticism of set theory, and its use mainly signals one's inclinations in that philosophical discussion.

The subject is otherwise generally known as set theory and has been known by that term for more than a century. It is a huge and vibrant field, with hundreds of active researchers and many exciting developments, some deeply connected with other parts of mathematics. Much of the terminology and even the notation of the subject has been remarkably stable since the time of Cantor, for if you take a peek into his "Contributions to the Founding of the Theory of Transfinite Numbers," you will find him, for example, using alpha and beta to refer to various countable ordinals, just as we often do today.

I recognize that the category theorists naturally seek a term to refer to this larger classical part of the subject, to distinguish it from the comparatively newer category-theoretic accounts of set theory, which I do view as welcome developments (and do not get me wrong on that point). But in such a case one should select a name acceptable to all, rather than one that highlights a criticism, especially when that criticism is generally not much accepted by its targets. If the category theorists cannot call it set theory, then I suggest that they call it classical set theory.

I have sometimes heard the term ZFC-style set theory, but this is a little inaccurate, since set theorists also freely work in ZF or other much weaker set theories, such as ZFC- or Kripke-Platek, or in various second-order set theories, such as Gödel-Bernays set theory GBC or Kelley-Morse set theory KM. So ZFC isn't the full story.

Meanwhile, I resist the confusion in terminology that results from trying to appropriate the term set theory to refer mainly or only to the category-theoretic approach to set theory, which it seems may be the real intent of the attempted change in terminology. It seems wrong to me for critics to try to rename an established subject in a way that is rejected by the practioners, especially when the new term places undue emphasis on the criticism. I would find it similar to someone insisting on using the term pro-abortion instead of pro-choice or anti-choice instead of pro-life, a practice I find unlikely to lead to a fruitful intellectual exchange of ideas.

My own view is that the category-theoretic approach to set theory is a welcome development that can be viewed simply as a part of set theory, rather than as a replacement of it, a part particularly connected or overlapping with category theory. But meanwhile, there are many other parts of set theory, including descriptive set theory, set-theoretic topology, cardinal characteristics of the continuum, large cardinals, forcing and so on.

So please don't use the term material set theory, unless you particularly want to signal your non-neutrality in a philosophical dispute.

Comments welcome!

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Here comes Hilbert's Half-Marathon, with one runner for each non-negative rational number. Will they fit into Hilbert's Hotel? Into which room will you place runner p/q?

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A mathematical gem, showing that all triangles are isosceles. It follows that also every triangle is equilateral. Can you spot any error?

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I happened to ride the new 2nd Avenue subway line today, and was pleased to find the stations filled with dozens of beautiful intricate tile mozaics. Enjoy! 
17 Photos - View album

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The journal system is broken. 

Here is the story of the publication of a recent paper of mine:

October 2011: Research completed.
December 2011: Paper submitted to arXiv.
January 2012: Contacted by another researcher, who could improve our results. Revised the paper and added new author.
February 2012: Submitted to journal.
[9 months pass]
November 2012: Rejected by journal, received one 5-sentence referee report with no objective criticism.
January 2013: Submitted to a second journal (stronger than the first).
[9 months pass]
October 2013: Accepted by journal with minor revisions.
February 2014: Submit revised paper to journal (including source files).
April 2014: Journal acknowledges receipt and reconfirms acceptance.
[10 months pass]
February 2015: Authors enquire what's happening. Managing editor says he will investigate.
April 2015: After no further contact from journal, authors again enquire.
May 2015: Managing editor writes, "there has been a misunderstanding regarding your paper, the technical staff was under the impression that it is still under review" ! Authors wonder why this was not picked up by managing editor's investigation in February. Managing editor says to expect galley proof soon.
May 2015: Three weeks later, no galley proof. Authors write to editors-in-chief. No response.
June 2015: Authors write to managing editor again. Four days later, receive galley proofs. Two days later authors submit minor corrections to galley proofs.
August 2015: Paper published online.
[16 months pass]
December 2016: Paper published in print.

So what do I take from this?
(1) It says how important arXiv is. The results were significantly improved, and a new collaboration started because of arXiv. The arXiv paper was cited more than 20 times prior to journal publication.
(2) This history points to the insignificance of journal publication. The only people who care about prestigous journal publications are funding agencies and university bureaucrats.
(3) That the second journal sat on this paper for two years doing nothing highlights that commercial publishers are adding very little to the value of the paper. It is then extraordinary that this journal charges many thousands of dollars in subscription fees.
(4) The good news is that this paper has counted towards my "current work" for many years more than it deserves :-)

Consider any statement, such as, `You will find eternal happiness.' I shall prove that it is true. In order to do so, let me first introduce three propositions:

Proposition 1 is the assertion: This proposition has the opposite truth value of the next one.

Proposition 2 is the assertion: This proposition has the same truth value as the next one.

Proposition 3 is the assertion: You will find eternal happiness.

Now, if proposition 1 is true, then it follows that proposition 2 is false, in which case proposition 3 must be true, so you will find eternal happiness. Otherwise, proposition 1 is not true, in which case, proposition 2 is true, from which it follows that proposition 3 is also true. So you will find eternal happiness in any case.

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A new undergraduate course I've been developing: Introduction to proofs.

I've been writing a book on the topic: Proof and the art of mathematical reasoning. 

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What do we want? Evidence-based science.
When do we want it? After peer review.

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The gamut of mathematical emotions!

Making numbers from 2 0 1 7. My daughter (10 years old) was given the task by her math teacher to form as many numbers as she could using the numbers: 2, 0, 1, 7, exactly once each, and the operations of addition, subtraction, multiplication, division, exponents, and factorial, with free use of parentheses. (It is not allowed to combine the numbers directly as digits, like 21, but you can make 21 as 7*(2+1)+0.)

0 = 0*(2^(7+2+1)!!)
1 = 2-1+0*7
2 = 2+0*(7+1)
3 = 2+1+0*7
and so on...

She went a long way, and made a lot of numbers. But there is a gap in her list: she's couldn't make 19.

Question. Can you make 19? How to prove that this is impossible?

I guess we should give ourselves sqrt, if this helps, or something else?

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