As to 1. MU was to be an ideographic language in which graphical signs would be directly coordinated with (concepts) ideas, i.e. without any mediating of an ordinary spoken language (as was e.g. in Aristotle's logic).
As to 2. The language of arithmetic has its own syntax independent of natural langages grammar, what gives it a perfect simplicity and perspicuity, much capable of guiding mind in reasonings (like Ariadne's thread -- in a favourite Leibniz's parable).
As to 3. "Pure thought" denotes the thought as an objective abstract entity, defined independently of any subjective mental states of individuals, like in a mathametical language (again, in accordance with MU intentions).

All these features of Frege's work are essential for each system of contemporary symbolic logic, and this is a way in which symbolic logic may be deemed as MU revisited. With examining how is MU related to Cantor's settheoretical ideas, we shall make a next step towards the informational worldview -- as a modern embodiment of MU vision.

§3. Cantor's ingenious contribution to lingua universalis

Georg Cantor's (1845-1918) role in creating the modern MU version is tremendous. While Frege devises a conceptual notation and an efficient calculus for reasoning (Leibniz's calculus ratiocinator) as one of the two MU modules, Cantor provides us ingeniously with the other module: a language based on some primitive notions which are to be basic not only for the language of mathematics but the whole knowledge as well. Such a universal language was by Leibniz named in many ways (see Kluge quoted in §2, p.234); among all of them the most fitting appears to be lingua universalis.

These primitive notions play the key role in drawing the borderline between the finite and the infinite. This distinction entered the basics of informational worldview. It displays a crucial difference between the mechanical and the creative way of processing information in a problem-solving behaviour.

Leibniz's considered a number of concepts for the role of conceptual primitives (called also semantic universals). However, he failed to notice that it is the notion of set (or class, in an alternative vocabulary) which should be the most favourite candidate. Its meaning gets characterized with a set of axioms, and then (with the help of some logical terms) serves as basis to introduce other notions, including that of equinumerosity or (another term) equipollence, having a great import on developing the universal language.

Two sets are said to be equinumerous if there is one-to-one correspondence between their elements (as, say, between fingers of left and right hand). A composition of these two ideas results in defining number (more precisely, cardinal number) as something what any two equipollent sets have in common (say, having five elements as feature shared by sets of hand fingers). Thus the Leibnizian strategy -- of composing more complex notions out from simpler ones -- gets realized. According to the same strategy, so defined idea of cardinal number becomes the basis to successively introduce more composite numbers (negative, rational, real etc).

This process of step by step developing a universal language, based on the notion of set at the very bottom, is one "whose importance for the beginning of civilization in general can hardly be over-estimated" -- wrote Abraham Fraenkel, one of the greatest contributors to set theory in his Abstract Set Theory (North Holland, Amsterdam etc. 1976, see p.24). This Fraenkel's view on civilization ought to be counted among main tenets of informational worldview.

The notion of equinumerosity plays a decisive role in making us aware of singularities of infinite sets. A set is infinite then and only then when it has a proper part equinumerous with the whole set; say the set of prime numbers being equinumerous with the set of all natural numbers in which it is included, i.e., forms a subset. However, not every infinite set having an infinite subset is equinumerous with it. Cantor was the first who noticed and proved the astonishing fact that the infinite set of natural numbers, being a subset of infinite set of reals, does not equal the latter: there are more real numbers than natural numbers.

Now, to render the relation between MU as a legacy handed over to us by earlier ages, and its modern realization in our time, let us observe the following.

Some items of the original project which for the lack of relevant devices could not be carried out by its originators, have been completed with modern logic. We have got the conceptual language of symbolic logic, and very efficient calculus for deduction, as well as the method of successiwe introducing more complex notions on the basis of simpler primitives. The Cantorian notion of set as the fundamental primitive idea, symbolic languages of Frege, but also those of Peano, Russell, Hilbert, Łukasiewicz etc., exemplify the accomplishment of characteristica universalis, while the logical calculus initiated by Frege, and developed in various versions by those mentioned above, forms a fulfillment of the Leibnizian dream of calculus ratiocinator. Moreover, such calculi are being performed effectively by machines, as Leibniz prophetized to come.

However, the idea of universal mathematically shaped knowledge must have deeply changed in the face of dramatic changes in our picture of the world. The universe as seen by modern people is no longer a closed static reality. Instead, it appears to us as an open and dynamic system tending to ever greater complexity. This understading started with the discovery of the universe's evolution as reflected in Einstein's general relativity, hence its significance for the modern MU. While exploring ever new realms of the universe, that display a growing complexity, we need more and more advanced mathematical models, as exemplified with the history of relativity and quanta. This continously moves the frontiers of modern MU ahead, and shows that it is not likely to reach whenever a final stage.

On the other hand, Gödel teaches us optimistically that the dynamics of human mind is able to keep up with the dynamically growing complexity of research problems. This is a corrolary to Gödel's statement that at any phase of the development of arithmetic there are problems not solvable in it in a mechanical way, but the mind's creative intuition can devise fresh axioms and rules to create new means of mechanical problem-solving. However, when the word "intuition" appears, one may wonder if we remain true to the very MU idea, since MU should have provide us with means of solving any problem whatever in a computational ("calculesmu"), that is, mechanical manner.