[Footnotes to Part 1]
[1] Cf. the summary of the results of this work, published in Anzeiger der Akad. d. Wiss. in Wien (math.-naturw. Kl.) 1930, No. 19.
[2] A. Whitehead and B. Russell, Principia Mathematica, 2nd edition, Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), and the axioms of reducibility and of choice (for all types).
[3] Cf. A. Fraenkel, ‘Zehn Vorlesungen über die Grundlegung der Mengenlehre’, Wissensch. u. Hyp., Vol. XXXI;
J. v. Neumann, ‘Die Axiomatisierung der Mengenlehre’, Math. Zeitschr. 27, 1928, Journ. f. reine u. angew. Math. 154 (1925), 160 (1929).
We may note that in order to complete the formalization, the axioms and rules of inference of the logical calculus must be added to the axioms of set-theory as given in the above-mentioned papers. The remarks that follow also apply to the formal systems presented in recent years by D. Hilbert and his colleagues (so far as these have yet been published).
Cf. D. Hilbert, Math. Ann. 88, Abh. aus d. math. Sem. der Univ. Hamburg I (1922), VI (1928);
P. Bernays, Math. Ann. 90;
J. v. Neumann, Math. Zeitsehr. 26 (1927);
W. Ackermann, Math. Ann. 93.
[4] i.e: more precisely, there are undecidable propositions in which, besides the logical constants ¬ (not), ∨ (or), (x) (for all) and = (identical with), there are no other concepts beyond + (addition) and . (multiplication), both referring to natural numbers, and where the prefixes (x) can also refer only to natural numbers.
[5] In this connection, such axioms in PM are only counted as distinct axioms where they do not arise from others purely by change of type.
[6] Here and in what follows, we shall always understand the term “formula of PM” to mean a formula written without abbreviations (i.e: without use of definitions). Definitions serve only to abridge the written text and are therefore in principle superfluous.
[7] i.e: we map the basic symbols by a one-to-one correspondence onto the natural numbers (as is actually done in the section [The Gödel numbering system]).
[8] i.e: a mapping of a section of the number series onto natural numbers. (Numbers cannot in fact be put into a spatial order.)
[9] In other words, the above-described procedure provides an isomorphic image of the system PM in the domain of arithmetic, and all meta-mathematical arguments can equally well be conducted in this isomorphic image. This occurs in the following outline proof, i.e: “formula”, “proposition”, “variable”, etc. are always to be understood as the corresponding objects of the isomorphic image.
[10] It would be very simple (though rather laborious) actually to write out this formula.
[11] Perhaps according to the increasing sums of their terms and, for equal sums, in alphabetical order.
[11a] The ¬ indicates negation. [The original German used a bar over a formula to indicate negation]
[12] Again there is not the slightest difficulty in actually writing out the formula S.
[13] Note that “[R(q); q]” (or – what amounts to the same thing – “[S; q]”) is merely a meta-mathematical description of the undecidable proposition. But as soon as one has ascertained the formula S, one can naturally also determine the number q, and thereby actually write out the undecidable proposition itself.
[14] Every epistemological antinomy can likewise be used for a similar undecidability proof.
[15] In spite of appearances, there is nothing circular about such a proposition, since it begins by asserting the unprovability of a completely determinate formula (namely the qth in the alphabetical arrangement with a definite substitution), and only subsequently (and in some way by accident) does it emerge that this formula is precisely that by which the proposition was itself expressed.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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