**Torkel Franzén** (1 April 1950, Norrbotten County – 19 April 2006, Stockholm) was a Swedish academic.

Franzén worked at the Department of Computer Science and Electrical Engineering at Luleå University of Technology, Sweden, in the fields of mathematical logic and computer science. He was known for his work on Gödel's incompleteness theorems and for his contributions to Usenet.^{ [1] } He was active in the online science fiction fan community, and even issued his own electronic fanzine *Frotz* on his fiftieth birthday.^{ [2] } He died of bone cancer at age 56.^{ [3] }

*Gödel's Theorem: An Incomplete Guide to its Use and Abuse*. Wellesley, Massachusetts: A K Peters, Ltd., 2005. x + 172 pp. ISBN 1-56881-238-8.*Inexhaustibility: A Non-Exhaustive Treatment*. Wellesley, Massachusetts: A K Peters, Ltd., 2004. Lecture Notes in Logic, #16, Association for Symbolic Logic. ISBN 1-56881-174-8.- The Popular Impact of Gödel's Incompleteness Theorem,
*Notices of the American Mathematical Society*,**53**, #4 (April 2006), pp. 440–443. *Provability and Truth*(Acta universitatis stockholmiensis, Stockholm Studies in Philosophy 9) (1987) ISBN 91-22-01158-7

In computability theory, the **Church–Turing thesis** is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:

**Gregory John Chaitin** is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy.

**Kurt Friedrich Gödel** was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Gottlob Frege.

**Mathematical logic** is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

In logic and deductive reasoning, an argument is **sound** if it is both valid in form and its premises are true. Soundness has a related meaning in mathematical logic, wherein a formal system of logic is sound if and only if every well-formed formula that can be proven in the system is logically valid with respect to the logical semantics of the system.

**Stephen Cole Kleene** was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene's work grounds the study of computable functions. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star, Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism.

In mathematics, a **theorem** is a statement that has been proved, or can be proved. The *proof* of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

**Gödel's incompleteness theorems** are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

In mathematics, **Hilbert's second problem** was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in Hilbert (1900), which include a second order completeness axiom.

**Proof theory** is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

**Metamathematics** is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.

**George Stephen Boolos** was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.

**Elwyn Ralph Berlekamp** was a professor of mathematics and computer science at the University of California, Berkeley. Berlekamp was widely known for his work in computer science, coding theory and combinatorial game theory.

**Johan Torkel Håstad** is a Swedish theoretical computer scientist most known for his work on computational complexity theory. He was the recipient of the Gödel Prize in 1994 and 2011 and the ACM Doctoral Dissertation Award in 1986, among other prizes. He has been a professor in theoretical computer science at KTH Royal Institute of Technology in Stockholm, Sweden since 1988, becoming a full professor in 1992. He is a member of the Royal Swedish Academy of Sciences since 2001.

"**Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I**" is a paper in mathematical logic by Kurt Gödel. Submitted November 17, 1930, it was originally published in German in the 1931 volume of *Monatshefte für Mathematik und Physik.* Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper contains Gödel's incompleteness theorems, now fundamental results in logic that have many implications for consistency proofs in mathematics. The paper is also known for introducing new techniques that Gödel invented to prove the incompleteness theorems.

In mathematics, a **proof of impossibility** is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a **negative proof**, **proof of an impossibility theorem**, or **negative result**. Proofs of impossibility often are the resolutions to decades or centuries of work attempting to find a solution, eventually proving that there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic.

**Georg Kreisel** FRS was an Austrian-born mathematical logician who studied and worked in the United Kingdom and America.

The **history of the Church–Turing thesis** ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable. It is an important topic in modern mathematical theory and computer science, particularly associated with the work of Alonzo Church and Alan Turing.

**John W. Dawson Jr.** is an American academic who is an emeritus professor of mathematics at Penn State York.

The **Penrose–Lucas argument** is a logical argument partially based on a theory developed by mathematician and logician Kurt Gödel. In 1931, he proved that every effectively generated theory capable of proving basic arithmetic either fails to be consistent or fails to be complete. Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing Machine that works on Peano arithmetic because the latter cannot see the truth value of its Gödel sentence, while human minds can. Mathematician Roger Penrose modified the argument in his first book on consciousness, *The Emperor's New Mind* (1989), where he used it to provide the basis of his theory of consciousness: orchestrated objective reduction.

- ↑ In Memory Of, web page at the American Mathematical Society, accessed August 2, 2007.
- ↑ "Frotz: An Electronic Oneshot". Archived from the original on October 1, 2007. Retrieved 2007-10-01., accessed online September 8, 2007.
- ↑ Torkel Franzén is dead, 20 April 2006.

- Home page
- Raatikainen, Panu. Review of
*Gödel's Theorem: An Incomplete Guide to Its Use and Abuse*.*Notices of the American Mathematical Society*, Vol. 54, No. 3 (March 2007), pp. 380–3.

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