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Hechler forcing is described on page 278, Jech.

Does anyone know when Hechler forcing was first used in a publication?

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Incidentally, Erin, Hechler sometimes shows up at our seminar, although it has been a while since I've seen him. –  Joel David Hamkins Apr 16 at 20:19
    
That's awesome, I hope I get to talk to him someday. –  Erin Carmody Apr 17 at 0:31

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The citation is given on page 283 of Jech (at least, in the 3rd Millenium edition); it is to Hechler's paper "On the existence of certain cofinal subsets of $^\omega\omega$," in the collection "Axiomatic Set Theory II" (see http://www.ams.org/books/pspum/013.2/9987/pspum9987.pdf), which is the only paper by Hechler in Jech's bibliography. It is somewhat difficult to read, but I think that we call "Hechler forcing" is a simplified version of the forcing introduced on page 159 of that paper; it seems reasonable to assume that this is its first instance.

(An historical note: Jech cites Hechler's paper as "[1974]," which is the date the collection "Axiomatic Set Theory II" was published; but that collection is the proceedings of a symposium held in 1967.)

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Thank you Noah. I just now was about to post that I had found it in Jech. And, the paper is very helpful. –  Erin Carmody Apr 16 at 19:51
    
@Erin: As a rule of thumb, it's usually a good idea to check Jech's book[s] and Kanamori's book[s] before asking. It's usually where the answer is going to come from anyway. –  Asaf Karagila Apr 16 at 20:23
    
@AsafKaragila, when I was a graduate student, Adrian Mathias and I were once discussing some set-theoretic matter, and when I told told him that I thought it was in Jech's book, he told me that he tried never to look things up there, since to do so, he felt, meant that he had failed himself. And indeed, Adrian figured it out what we were talking about just then. (But of course citation and reference information are a different matter...) –  Joel David Hamkins Apr 16 at 21:40
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Oh, I think the attitude is a very good one! For my part, I'm glad to have picked it up from Adrian. The point is that one learns things more deeply when figuring it out on one's own, and surprisingly often, one finds a better way to do things. This is especially important when one is just starting on a new topic, which is not yet completed in the other accounts, since otherwise one can get trapped in someone else's unproductive perspective. –  Joel David Hamkins Apr 16 at 23:19
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In connection with "somewhat difficult to read", it may be worth reminding people that the 1967 symposium was, I believe, where both Shoenfield's unramified forcing and Scott and Solovay's Boolean-valued models were first presented. Before then, forcing was usually done in the style of Cohen's original work, using a ramified language of terms in a forcing extension of $L$, using strong rather than weak forcing (strong forcing doesn't respect classical logic), etc. No wonder it was difficult to read. (Young people nowadays don't realize how good they have it.) –  Andreas Blass Apr 28 at 0:31

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