Talk:Epsilon calculus

Bourbaki notation

From the article:

This notation is equivalent to the Hilbert notation and is read the same.

Really? $(\tau _{x}(A)|x)A$ is pronounced "epsilon eks ey"? I find that hard to believe.—Emil J. 15:47, 3 December 2010 (UTC)

Good question! The point is that anything you formulate using the $\varepsilon$ Hilbert style, you can formulate an equivalent statement using the Bourbaki $\square -\tau$ style.

"Equivalent" here does not mean "$\varepsilon =\tau$ and ...", but it means that there is an effective algorithm to translate notations in finitely many steps.

Hope that helps...—Pqnelson (talk) 16:20, 28 October 2011 (UTC)

Actually, I found this explanation on mathoverflow a 4-step algorithm to translate the $\varepsilon$ notation to Bourbaki's notation!

(1) Replace this $\varepsilon$ with $\tau$.

(2) Erase the variable that comes right after the $\varepsilon$.

(3) Replace all subsequent occurrences of that variable with a box.

(4) Link each of those boxes to the $\tau$ you wrote in (1). So $(\varepsilon x)\phi (x)$ becomes $\tau \phi (\square )$ with a link from the $\tau$ to the boxes (as many boxes as there were $x$'s in $\phi (x)$).

Of course, this is a bit sloppy, but negligibly so. To make it more rigorous, you need to modify these steps to loop over the variables $x,y,z,...$. But that's implicit. —Pqnelson (talk) 22:39, 28 October 2011 (UTC)

Let's work on our explanation

Can someone please explain to me how we might paraphrase $\ A(\epsilon x\ (\neg A))$ in natural language, in a way that makes it obvious how it is equivalent to $(\forall x)A(x)\$? Afterwards we can include that explanation in the main article. Thank you. :) --77.204.222.12 (talk) 22:00, 4 December 2010 (UTC)

Do you understand why $\exists x\,B(x)$ is equivalent to $B(\epsilon x\,B(x))$? Then take $B=\neg A$: $\forall x\,A(x)\Leftrightarrow \neg \exists x\,\neg A(x)\Leftrightarrow \neg \neg A(\epsilon x\,\neg A(x))\Leftrightarrow A(\epsilon x\,\neg A(x))$.—Emil J. 15:24, 6 December 2010 (UTC)

Explain the notation to a reader

This article assumes that a reader who comes to this page understands what the notation means. It would be extremely useful to the reader of this and other mathematical articles if the mathematical notation is explained step by step, otherwise it just remains abstruse going over their (mine too) heads. If the paragraph under the image of the notation is supposed to be an explanation then it is not doing the job in my opinion.Chandraputra (talk) 08:29, 9 November 2014 (UTC)

Relation to formal languages?

The relation to formal languages is quite foggy, if not totally unclear to me. There is a section in the article on formal languages about how they are used to define a language for well-formed formulas and theorems, and I suppose it is in this regard that epsilon calculus works. But there could well be some more and better explanation. --Lasse Hillerøe Petersen (talk) 18:30, 24 April 2017 (UTC)