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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.

²⁸already cited