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35MM FILM LEADER WITH TEST PATTERN #1216/2009
(  Satoshi Kinoshita )
| Series: | Prints on paper: 35mm Film Leader | | Medium: | Giclée on Japanese matte paper | | Size (inches): | 16.5 x 11.7 (paper size) | | Size (mm): | 420 x 297 (paper size) | | Edition size: | 25 | | Catalog #: | PP_0174 | | Description: | From an edition of 25. Signed, titled, date, copyright, edition in pencil on the reverse / Aside from the numbered edition of 5 artist's proofs and 2 printer's proofs.
"The development of mathematics towards greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules." - Kurt Gödel, opening of the paper introducing the Undecidability theorem (1931).
-http://www-history.mcs.st-and.ac.uk/Quotations/Godel.html
Gödel's first incompleteness theorem states that:
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250).
Gödel's second incompleteness theorem can be stated as follows:
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.
-http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems
Kurt Gödel -
Kurt Gödel (German pronunciation: [ˈkʊʁt ˈɡřːdəl]; April 28, 1906, Brno, Moravia, Austria–Hungary – January 14, 1978, Princeton, New Jersey, United States) was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.
Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
The Incompleteness Theorem:
In 1931 and while still in Vienna, Gödel published his incompleteness theorems in "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (called in English "On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:
1. If the system is consistent, it cannot be complete.
2. The consistency of the axioms cannot be proven within the system.
These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.
In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to solve several technical issues, such as encoding statements, proofs, and the very concept of provability into the natural numbers. He did this using a process known as Gödel numbering.
In his two-page paper "Zum intuitionistischen Aussagenkalkül" (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).
-http://en.wikipedia.org/wiki/Kurt_Gödel
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