1 “Computable” economics is to be distinguished from “computational” economics, with the former focusing on fundamental issues such as the computability or undecidability of certain systems, whereas the latter tends to be more concerned with more superficial technical matters such as how to get particular programs to run more quickly. The latter also studies simulations of particular economic models.
2 A simple statement of this axiom due to Rosser (, p. 88) is “If λ is a set of nonempty, nonoverlapping sets, then there is a set γ which has exactly one member in common with each member of λ.” Given that λ can vary, which is usually a transfinite cardinal, there are many versions of the axiom, with Zorn’s Lemma  usually being assumed to assert it holds for most levels that most mathematicians deal with, although Specker  showed that it does not hold at the ultimate level of the universe as a whole, using the law of the excluded middle, or reductio ad absurdum, thus rendering absurd a constructivist assertion of this theorem as a general disproof of the axiom. Specker’s result uses the Quine’s New Foundations approach.
3 “Simple consistency” is Rosser’s terminology, whereas some others call this “ordinary consistency” and Gödel simply called it “consistency.” It was Gödel who introduced the concept of ω-consistency. Both of these papers, along with important ones by Alonzo Church, Alan M. Turing, Stephen C. Kleene, and Emil Post, can be found in Davis .
4 A traditional method to escape the paradoxes of Russell, including the famous one regarding whether the set of all sets that are not members of themselves is a member of itself, is to place restrictions on the use of certain names that lead to such paradoxes. It has been argued that many of the paradoxes of logic arise from problems related to “naming the names” of mathematical objects.
5 Velupillai also notes that there is a great similarity between the constructive approach to proving the Hahn-Banach Theorem and the Intermediate Value Theorem, which in turn has indirect links with the Brouwer Fixed Point Theorem. However, we shall not consider Vellupillai’s discusson of the intermediate value theorem further in this paper, as it is not tied to any specific economic application, nor does it involve any further mathematical concepts beyond what we deal with in the Hahn-Banach and Bolzano-Weierstrass theorems. I thank Eric Bach for pointing out to me that the finite-dimensional version of the Hahn-Banach Theorem can be proven without any recourse to the Axiom of Choice or its weaker ultrafilter relatives.
6 The pieces of a Banach-Tarski decomposition must be non-measurable; therefore by the celebrated result of Robert Solovay  such decompositions cannot be built without some use of the Axiom of Choice and hence lie outside constructive mathematics, suggesting that perhaps Velupillai is right to put his tongue into his cheek when referring to them. (I thank Adrian Mathias for clarifying this matter for me.)
7 Brouwer  did eventually follow up to provide a correction to his proof to make it compatible with intuitionism.
8 Often it is Sperner’s lemma that is invoked, but this depends on the Bolzano-Weierstrass theorem for its proof. See Tompkins  for further discussion.
9 Needless to say, there are economists who find showing how difficult it is to prove convergence to a fixed point to be quite uninteresting in the face of experimental evidence that at least in experimental double auction markets most agents are able to move to (partial) equilibrium solutions quite rapidly with little thought .
10 While most western works on Zen Buddhism emphasize this view of the koans, there is another more “classical” approach in which they have specific answers that the adept is expected to learn and repeat in a nearly rote way, once learned. Hoffman  provides a “cheat sheet” of answers for some of the more famous ones, with, for example, the official answer for the most famous one of all, “what is the sound of one hand clapping?” being simply the act by the adept to stand correctly before the master and to thrust his hand forward decisively. Thinking of this sort of Zen leads to understanding how it could be related to an authoritarian use of martial arts.
11 This is an interpretation that Gödel  appears to have accepted. However, others distinguish more sharply between “many-valued logics” that assign definite truth values (interpretable as probabilities of truth), and intuitionism, which is seen as less definite on such truth values by Kleene  as well as Rosser and Turquette . More recently, Atanassov  has proposed a combination of the approaches in the theory of “intuitionistic fuzzy sets.”
12 The classical and rigorous approach to mathematical physics is codified in Courant and Hilbert .
13 I thank David Levy for pointing out to me that almost certainly whatever ambiguities Newton may have entertained regarding the nature of his fluxions, they were not due to Bishop Berkeley’s jibes against them as the “ghosts of departed quantities,” as Berkeley did not make his arguments until 1734 in his The Analyst, whereas Newton died in 1727.
14 In this joint paper from 1935 , they showed that the λ-calculus is inconsistent, although modified versions of it have since been proven to be consistent. They were among the developers of this system, much used later by computer scientists and artificial intelligence programmers. Others closely involved in its initial development were Haskell Curry and Alonzo Church, the latter the major professor of both Kleene and Rosser.
15 It may be that now the most intensively studied of his theorems is the one he proved with Church that shows the “diamond property” of recursive systems, the Church-Rosser theorem , which was very pure mathematics when they proved it prior to the invention of computers with memory, but is now viewed as deeply practical by computer scientists.
16 Like his friend, John von Neumann, he continued to be involved with the military and intelligence after that war, receiving numerous commendations for his mostly classified work, including one for his solution of the water-to-air phase transition problem of the first submarine-launched missile, the Polaris. Later, during the Vietnam War this work of his would become a matter of public controversy .
17 In 1929 he forecast to a group of highly skeptical friends that humans would land on the moon within 50 years. When this event occurred in 1969, he said that what he had not forecast was that he would be able to view it live on television (which he rarely watched otherwise, mostly considering it a “waste of time”).
18 He played a crucial role in the first successful manned flight to the moon. He solved a problem of astronauts landing on earth further from their planned rendezvous sites with longer flights as due to a discrepancy between the clocks on the ground and those in space, with the former on solar time and the latter on sidereal (star) time, off from each other by the roughly 1/365 due to the annual revolution of the earth about the sun. If this had not been resolved, the first flight to the moon would have gone into deep space without hope of return.
19 While he often enjoyed confounding others, he could also be reassuring when it suited him. Thus, during a public presentation, a young woman asked him if zero was a real number. Ever the perfect southern gentleman, he replied, “One of the finest, my dear, one of the finest.”
20 Adrian Mathias has pointed out to me that more properly speaking the Boolean valued model approach to forcing assigns truth values in a complete Boolean algebra to sentences of the forcing language. This may have nothing to do with probabilities of truthfulness per se.
21 However, Atanassov  has more recently developed an intuitionistic extension of fuzzy set logic. See also .
22 A sideshow to this discussion involves theology. Does believing in an ideal, Platonic, mathematical reality imply that one must believe in a deity? Hardy is proof that this is not necessarily the case, as he was both a publicly confirmed and vigorous atheist, while also accepting that mathematical reality is a Platonic ideal outside of us. As for my father’s theological views, I will note that while he attended church regularly with my very religious mother, he never definitively stated his position on these matters in my hearing, and I am less certain of his ultimate position regarding them than I am about what his political views were.
23 I thank Jerome Keisler for informing me that the understanding of this consistency and the tradeoffs involved has now proceeded very far.