on 03-Oct-2020 (Sat)

Consistency (internal): when every theorem, upon interpretation, comes out true (in some imaginable world)

So far, the only way we have found to represent prime numbers typographically is as a negative space. Is there, however, some way—I don't care how complicated—of representing the primes as a positive space—that is, as a set of theorems of some formal system? Different people's intuitions give different answers here. I remember quite vividly how puzzled and intrigued I was upon realizing the difference between a positive characterization and a negative characterization. I was quite convinced that not only the primes, but any set of numbers which could be represented negatively, could also be represented positively. The intuition underlying my belief is represented by the question: "How coulda figure andits ground not carry exactly the same information?" They seemed to me to embody the same information, just coded in two complementary ways. What seems right to you? It turns out I was right about the primes, but wrong in general. This astonished me, and continues to astonish me even today. It is a fact that: There exist formal systems whose negative space (set of non- theorems) is not the positive space (set of theorems) of any formal system

these are my own terms—they are not in common usage). A cursively drawable figure is one whose ground is merely an accidental by-product of the drawing act. A recursive figure is one whose ground can be seen as a figure in its own right. Usually this is quite deliberate on the part of the artist. The "re" in "recursive" represents the fact that both foreground and background are cursively drawable—the figure is "twice-cursive".

For a set of strings to be recursively enumerable means that it can be generated according to typographical rules

There exist recursively enumerable sets which are not recursive. The phrase recursively enumerable (often abbreviated "r.e.") is the mathemat ical counterpart to our artistic notion of "cursively drawable"—and recursive is the counterpart of "recursive".

we saw how meaning—at least in the relatively simple context of formal systems—arises when there is an isomorphism between rule-governed symbols, and things in the real world

By treating words such as "POINT" and "LINE" as if they had only the meaning instilled in them by the propositions in which they occur, we take a step towards complete formalization of geometry. This semiformal version still uses a lot of words in Englishwith their usual meanings (words such as "the", 'if", "and", "join", "have"), although the everyday meaning has been drained out of special words like "POINT" and "LINE", which are con sequently called undefined terms. Undefined terms, like the p and q of the pq-system, do get defined in a sense: implicitly—by the totality of all proposi tions in which they occur, rather than explicitly, in a definition.

A full formalization of geometry would take the drastic step of making every term undefined—that is, turning every term into a "meaningless" symbol of a formal system. I put quotes around "meaningless" because, as you know, the symbols automatically pick up passive meanings in accordance with the theorems they occur in.

We began our discussion by manufacturing what appeared to be an inconsistent formal system—one which was internally inconsistent, as well as inconsistent with the external world. But a moment later we took it all back, when we realized our error: that we had chosen unfortunate interpretations for the symbols. By changing the interpretations (in pq system, q changed its interpretation from "equal" to "equal or greater than"), we re gained consistency! It now becomes clearthat consistency isnota property ofa formal system per se, but depends on the interpretation which is proposed for it.

what is meant by consistency of a formal system (together with an interpretation): that every theorem, when interpreted, becomes a true statement

internal consistency depends upon consistency with the external world—only now, "the external world" is allowed to be any imaginable world

Completeness: when all statements which are true (in some imaginable world), and which can be expressed as well-formed strings of the system, are theorems.