Does anybody shed light on what is A. R. D. Mathias' idea that Bourbaki's $\tau$-calculus (Logically the same as Hilbert's $\varepsilon$-calculus) is not suitable for set theory, especially because of incompleteness of set theory? I have fully read Bourbaki's Theory of Sets and Mathias' two papers on bourbaki's system (1, 2), but still cannot find any "real" logic flaw.

Especially, Mathias mentioned that $\tau$-calculus works perfectly on completed system like models of set theory, but works problematically on incomplete system like ZFC. But why? He cited many early publications of Bourbaki members, in which they misused unprovable as false. But at least in 1970 version of Theory of Sets, there is no such misusing.

For my understanding, $\tau$-operator works like FIND function in computer programming languages. It will accept property and give a return value. If there is such an element satisfied the property, it will give one correct element. If no such element exists, it will return a certain element (it is still a real element, we cannot say it has no meaning at all.). The quantifier $\exists xR(x):=R(\tau_x(R))$ is actually the verification of the returned element.

Also, Mathias says in $\tau$-calculus, we cannot discuss the concept of proper classes because they become equal. But, as we known, in ZFC, proper classes are not legal objects and can only be discussed in metalogic.

What is the connection of incompleteness and inproperly use of $\tau$-calculus in set theory? Is it only because Mathias' dissatisfaction on Bourbaki members' early misunderstand of incompleteness giving a negative impact on Set Theorist?

they themselvescould do it if they wanted to). This does rely on a certain conceptual simplicity at all levels. $\endgroup$ – Noah Schweber Nov 17 '19 at 17:52