# Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?

A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already earlier for the combinatorial principle?

Is there some (deeper) reason explaining how $\diamondsuit$ got its name? Perhaps because $\Box$ was already established as a combinatorial principle?

• I've always assumed it came next after $\clubsuit$ – Hurkyl Jul 13 '15 at 11:50
• Asaf Karagila. The phrase "A diamond is forever" is simply a very successful slogan from the diamond industry to make people buy diamonds. Also, diamonds only appear rare because a certain organization has a near monopoly on diamonds and diamond mines and they want us to think they are rare. In reality, they have piles and piles of diamonds. – Joseph Van Name Jul 13 '15 at 17:16
• Avshalom. My source is Nicky Oppenheimer, the director of DeBeers and richest man in South Africa who made all of his money from diamonds, who said "Diamonds are intrinsically worthless, except for the deep psychological need they fill." The story I gave about diamonds is well known. – Joseph Van Name Jul 13 '15 at 19:42
• @Joseph: I am fully aware of this. Unfortunately, textual conversation does not always convey humor properly. The correct reply would have been that diamonds can be easily destroyed by adding many reals, so they can't be that precious, strong or forever. – Asaf Karagila Jul 14 '15 at 4:24
• @Avshalom: That's why I got into set theory. Fertile ground for awesome titles. :-P (Also, I was instantly good at set theory, and rarely good in anything else. But that's besides the point...) – Asaf Karagila Jul 14 '15 at 6:42

I once asked Jensen this question, when we were at a conference at Oberwolfach.

I told him that I had always assumed that the diamond $\Diamond$ principle was called $\Diamond$ because it expresses that there is an object exhibiting an elaborate degree of internal reflection. After all, if $\langle A_\alpha\mid\alpha<\omega_1\rangle$ is a $\Diamond$-sequence, then it very often happens for $\alpha<\beta$ that $A_\beta\cap\alpha=A_\alpha$, which is an instance of $A_\beta$ reflecting to $A_\alpha$.

But alas, this was not his reason. The actual reason was less interesting, having mainly to do, he said, with him simply needing a new symbol that had not yet been used, and that one was available.

Meanwhile, I believe that we should all adopt my explanation anyway! Henceforth, let it be known that a $\Diamond$ sequence is one exhibiting beautifully captivating internal reflections!

• I don't know. I like the original explanation. It's like the origin of the Ctrl-Alt-Delete combination. They just needed a way to reboot the system quickly, and that was an unused combination. – Asaf Karagila Jul 13 '15 at 14:32
• Just to amplify "available": Anyone who printed work in logic would have the diamond and square symbols available, because of their use in modal logic. Jensen probably expected that his use of these symbols for combinatorial principles would never clash with their use for modal operators. That was too optimistic; I once wrote a paper in which both uses of square occur. But I"m not aware of any clash involving diamond. – Andreas Blass Jul 13 '15 at 15:38
• In the modal logic of forcing, the assertion $\Diamond\Diamond$ is a theorem of ZFC, where the first diamond means "forceable" and the second is the combinatorial principle. Meanwhile, $\neg\Diamond\square\Diamond$ is also provable, which is to say that the combinatorial principle is not possibly necessary. – Joel David Hamkins Jul 13 '15 at 15:43
• That internal reflection principle sounds attractive but it is too weak. Is it possible to view a diamond sequence as some stronger sort of internal reflection principle? My formulations couldn't get past CH. – Ashutosh Jul 13 '15 at 16:00
• @Andreas: So you had a proof there you had $\square\square\square$, where the first one is a modal statement, the second is a combinatorial property, and the third one means "QED"? :-) – Asaf Karagila Jul 14 '15 at 4:26