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2The Logical Syntax of Language, p. 5.
3Paul Lorenzen emphasizes this in "Methodical Thinking," Ratio, June 1965. For more from Lorenzen, see Constructive Philosophy (1987).
4I don't want to adduce a sheaf of references to support this. It is enough to add Kurt Gödel, "Russell's Mathematical Logic," in The Philosophy of Bertrand Russell, ed. Paul A. Schilpp (1944), p. 137.
5K.G. Binmore, The Foundations of Analysis, Book 1, p. 5.
6F.P. Ramsey, The Foundations of Mathematics (1931), pp. 20-21.
7I introduced the notion of a Failure Theorem in "Anti-Mathematics" (1980).
8Cf. Michael Dummett, Elements of Intuitionism (1977), p.10.
9From Frege to Gödel, p. 460.
10Whittaker & Watson, A Course of Modern Analysis, p. 3.
11Brouwer's objective, infinitary solipsism was no help.
12The Logical Syntax of Language, page 6. Incidentally, this led me to propose moments of the person-world as propositions. Critical Notes on Personhood, Part V.
13Ibid., pp. 5-6.
14Karl Menger, Selected Papers (1979), especially pp. 53-7. On page 51, Menger notes that Sierpinski invented the method of defining a real number by an unsolved problem.
15Menger, Selected Papers, p. 57.
16Menger, Selected Papers, p. 55.
17Remark: In the classroom, Quine gave an example of a system with only one symbol. Information is coded as the number of times the symbol is repeated. But in this case more than ever, you have to be able to perform real-world enumeration to read the system.
18Cf. "The Flaws Underlying Beliefs," Blueprint for a Higher Civilization (1975).
19The last version of this exercise was called "A Refutation of Arithmetic" (1988).
20Again, as we are in the orbit of a negative universal result, the explanations here are too casual. They have to be supplemented with more rigorous treatments in other places, such as Io #41.
21published in Blueprint for a Higher Civilization.
22One could say that "semantics" came into concept art from the time of "Subjective Propositional Vibration" (acronym SPV). Blueprint for a Higher Civilization, p. 201.
23Speculative sidelight: compare Gentzen's "consistency proof."
24But Carnap told himself that he could pass to pure combinatorics.
25This is a notion supported by The Logical Syntax of Language, p. 27, which has the term 'directly derivable'.
*published in An Anthology, ed. La Monte Young; see both 1963 and 1970 editions.
26Blueprint for a Higher Civilization, pp. 87-94.
27Not until 1987 would I begin to make new concept-art pieces, whose framework-breaking features were far more focused and explicit.
29or derivation, if you wish