abstract: "Russell's logical paradox, formulated in terms of English adjectives, is considered as a convenient starting point for this discussion of incomputability. It is shown to be impossible, under a wide variety of circumstances, to program a function which will determine whether another function written in the same programming language will terminate. The theory of types is introduced in an attempt to evade the paradox. Finally, it is shown that any language containing conditionals and recursive function definitions, which is powerful enough to program its own interpreter, cannot be used to program its own terminates function."
Source: Web. 11 Jan. 2015. <http://www.cse.chalmers.se/edu/year/2009/course/TDA183/docs/HoareAllison.pdf>.
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Georg Cantor (1845 – 1918) was a mathematician who invented set theory and proved that the set of real numbers is larger than the set of natural numbers. Thus implying that there exist infinite sets of different sizes (there is more than one infinity). In addition, he defined the cardinal and ordinal numbers and their arithmetic. His diagonal argument is fundamental in the solution of the Halting Problem and the proof of Gödel's first incompleteness theorem.
Source: Georg Cantor. (n.d.). Retrieved December 18, 2014, from http://en.wikipedia.org/wiki/Georg_Cantor Copper, Silver, Gold: An Indestructible Metallic Alloy is described in GEB as follows:
Gebstadter, Egbert B. Copper, Silver, Gold: an Indestructible Metallic Alloy. Perth: Acidic Books, 1979. A formidable hodge-podge, turgid and confused -- yet remarkably similar to the present work. Professor Gebstadter's Shandean digressions include some excellent examples of indirect self-reference. Of particular interest is a reference in its well-annotated bibliography to an isomorphic, but imaginary, book. (According to GEB, the phrase "turgid and confused" was used about the works of Johann Sebastian Bach by his contemporaries.) Egbert B. Gebstadter (author of Copper, Silver, Gold: An Indestructible Metallic Alloy) is a fictional author who Douglas R. Hofstadter mentions in the indexes and text of his books. For each Hofstadter book, there is a corresponding Gebstadter book. Note that the fictional author's name is derived from "GEB", the abbreviation for Hofstadter's first book Godel, Escher, Bach: An Eternal Golden Braid; the letters appear in his last name, permuted in his first name, and permuted again in his initials. Source: Egbert B. Gebstadter. (n.d.). Retrieved December 18, 2014, from http://en.wikipedia.org/wiki/Egbert_B._Gebstadter The Goldberg Variations is a work for harpsichord, by Johann Sebastian Bach, consisting of an aria and a set of 30 variations. It was published in 1741 and is considered to be one of the most important examples of variation form. The Variations are named after Johann Gottlieb Goldberg, who may have been the first performer.
Source: (n.d.). Retrieved January 10, 2015, from http://en.wikipedia.org/wiki/Jabberwohttp://en.wikipedia.org/wiki/Goldberg_Variationscky |