More music and other things!

How much can I torture my dear readers? Well, seeing as there are four and a half of them, I will not be increasing the overall pain in the world by much! Today’s post is intended to serve only one purpose – to record for myself some things that I want to remember! And so, dear reader, if you’re not down with it, we’ve got two words for you! – actually three – bear with me!

First up, probably the best concert I attended this season – my sixth Vijay Siva! Marvelous it was! With R K Sriramkumar on the violin, Tiruchi Sankaran Sir on the mridangam, and B S Purushothaman on the kanjira. He seems to be at the peak of his musical form! The concert was total sowkhyam throughout, everything kept simple and pure, with no gimmickry at all! And what choice of ragas! Hard to see so many weighty ragas packed into one concert!

  1. vallabhanāyakasya – begaḍa.
  2. saṅgītaśāstrajñānamu – mukhāri. Nice niraval, swaram.
  3. ambikāyām abhayāmbikāyām – kedāram. Short alapana, and niraval in the kriti.
  4. mīnalocana brova – dhanyāśi. I thought this was the best piece of the concert. Beautiful alapana, but the hightlight was the niraval, swaram in kāmapālinī.
  5. jānakīpate – kharaharapriya. Imagine singing kharaharapriya for a filler between two main pieces!
  6. śrisubrahmaṇyāya namaste – kāmbhoji. Majestic! The main piece, with an ālāpanā that is still fresh in memory, elaorate niraval and swarams.
  7. tani āvartanam – tisra ekam. I don’t know enough to say anything knowledgeable about Tiruchi Sankaran Sir’s playing except that it was scintillating, with B S Purushothaman ably matching him.
  8. rāgam tānam pallavi – ṣaṇmukhapriya – tisra tripuṭa (2). Not much time could be spent on the ālāpanā, but the pallavi had all the standard elements, singing in four tempos, etc.
  9. ini enna pechu – sahānā
  10. viruttam – yamunākalyāṇi, maṇiraṅgu, jañjūṭi
  11. tiruppugazh – jañjūṭi
  12. harivāsarada – sindhubhairavi
  13. maṅgalam

The other thing I want to record is something I fought hard to figure out more than once, and promptly forgot every time! After the latest episode (three or four days back) I decided to record it somewhere. (And a nice place I have chosen! Let me see if I find this the next time I am stuck on the same problem!) It has to do with a Lemma in Kenneth Kunen’s Set Theory that is dismissed with a one line proof! The claim has to do with relativizations. (The relativization of a formula \varphi with respect to another formula M(x) is got by replacing all quantifiers \forall{x}\alpha in \varphi by \forall{x}(M(x) \supset \alpha) and by replacing all quantifiers \exists{x}\alpha in \varphi by \exists{x}(M(x) \wedge \alpha). )

The claim is that if M(x) is a formula with one free variable, and \varphi_1, \ldots, \varphi_n, \psi are sentences such that \varphi_1, \ldots, \varphi_n \vdash \psi (this notation stands for: there is a derivation (using the axioms and rules for first-order logic) of \psi from the \varphi_i as assumptions), then \exists{x}M(x), \varphi^M_1, \ldots, \varphi^M_n \vdash \psi^M (here \varphi^M_1 etc. means the relativization of the formulas with respect to M(x)).

I was trying to transform a derivation of the former to a derivation of the latter. It didn’t work! Then I realised that one has to work with derivations that consist only of sentences. Whenever the assumptions and conclusion are sentences, one can manage things so that only sentences are used in the derivation, no matter what sound and complete axiom system for first order logic one uses. Then it’s a routine matter to check that the relativization of each axiom holds and that the relativized versions of the rules are admissible. But … even after many attempts I couldn’t pin down exactly where the extra assumption \exists{x}M(x) was being used in the second derivation. Finally I did find it, and that is what I want to record here!

The troublesome axiom is this: \forall{x}\alpha(x) \supset \alpha[x:=y]. But since our original proof consists only of sentences, the axiom that actually gets used is a generalization, \forall{y}(\forall{x}\alpha(x) \supset \alpha[x:=y]). Now the relativized version of this is \forall{y}(M(y) \supset ((\forall{x}(M(x) \supset \alpha(x)) \supset \alpha[x:=y])). One can check easily that this is a validity, and hence provable (or construct a derivation directly!).

So where is the extra non-emptiness assumption \exists{x}M(x) used? Precisely in proving the relativization of the above axiom, once we realize our error! The point is that x might not occur free in \alpha, and so \alpha[x:=y] might just be \alpha. In this case, our original derivation might have just used the axiom \forall{x}\alpha \supset \alpha. The relativized version is \forall{x}(M(x)  \supset \alpha) \supset \alpha. And this is not necessarily provable, unless we also assume “nonemptiness” of M! The antecedent says that for every x, either \neg{}M(x) holds or \alpha holds. Now if it is the case that for every x, \neg{}M(x) holds, then it is not necessary for \alpha to hold. Precisely this is ruled out by the assumption \exists{x}M(x).

Now I can go back to doing some actual work, after having disposed of this irritating gap in my understanding!

Update In my focus on the Kunen lemma, I forgot to mention another highlight of the concert. Padma and her father were there for this, too! And it was great company!

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