In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by a much simpler proof than that given below, since in addition to subsets of A with just one member, there are others as well, and since n < 2ⁿ for all natural numbers n. But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite. The theorem is named for German mathematician Georg Cantor, who first stated and proved it.

Proof[link]

Two sets are equinumerous (have the same cardinality) if and only if there is a one-to-one correspondence between them. To establish Cantor's theorem it is enough to show that, for any given set A, no function f from A into Failed to parse (Missing texvc executable; please see math/README to configure.): P(A) , the power set of A, can be surjective, i.e. to show the existence of at least one subset of A that is not an element of the image of A under f. Such a subset, Failed to parse (Missing texvc executable; please see math/README to configure.): B \in P(A) , is given by the following construction:

Failed to parse (Missing texvc executable; please see math/README to configure.): B=\left\{\,x\in A : x\not\in f(x)\,\right\}.

This means, by definition, that for all x in A, x ∈ B if and only if x ∉ f(x). For all x the sets B and f(x) cannot be the same because B was constructed from elements of A whose images (under f) did not include themselves. More specifically, consider any x ∈ A, then either x ∈ f(x) or x ∉ f(x). In the former case, f(x) cannot equal B because x ∈ f(x) by assumption and x ∉ B by the construction of B. In the latter case, f(x) cannot equal B because x ∉ f(x) by assumption and x ∈ B by the construction of B.

Thus there is no x such that f(x) = B; in other words, B is not in the image of f. Because B is in the power set of A, the power set of A has a greater cardinality than A itself.

Another way to think of the proof is that B, empty or non-empty, is always in the power set of A. For f to be onto, some element of A must map to B. But that leads to a contradiction: no element of B can map to B because that would contradict the criterion of membership in B, thus the element mapping to B must not be an element of B meaning that it satisfies the criterion for membership in B, another contradiction. So the assumption that an element of A maps to B must be false; and f can not be onto.

Because of the double occurrence of x in the expression "x ∉ f(x)", this is a diagonal argument.

[edit] A detailed explanation of the proof when X is countably infinite

To get a handle on the proof, let's examine it for the specific case when X is countably infinite. Without loss of generality, we may take X = N = {1, 2, 3,...}, the set of natural numbers.

Suppose that N is bijective with its power set P(N). Let us see a sample of what P(N) looks like:

Failed to parse (Missing texvc executable; please see math/README to configure.): P(\mathbb{N})=\{\varnothing,\{1, 2\}, \{1, 2, 3\}, \{4\}, \{1, 5\}, \{3, 4, 6\}, \{2, 4, 6,\dots\},\dots\}.

P(N) contains infinite subsets of N, e.g. the set of all even numbers {2, 4, 6,...}, as well as the empty set.

Now that we have a handle on what the elements of P(N) look like, let us attempt to pair off each element of N with each element of P(N) to show that these infinite sets are bijective. In other words, we will attempt to pair off each element of N with an element from the infinite set P(N), so that no element from either infinite set remains unpaired. Such an attempt to pair elements would look like this:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N}\begin{Bmatrix} 1 & \Longleftrightarrow & \{4, 5\}\\ 2 & \Longleftrightarrow & \{1, 2, 3\} \\ 3 & \Longleftrightarrow & \{4, 5, 6\} \\ 4 & \Longleftrightarrow & \{1, 3, 5\} \\ \vdots & \vdots & \vdots \end{Bmatrix}P(\mathbb{N}).

Given such a pairing, some natural numbers are paired with subsets that contain the very same number. For instance, in our example the number 2 is paired with the subset {1, 2, 3}, which contains 2 as a member. Let us call such numbers selfish. Other natural numbers are paired with subsets that do not contain them. For instance, in our example the number 1 is paired with the subset {4, 5}, which does not contain the number 1. Call these numbers non-selfish. Likewise, 3 and 4 are non-selfish.

Using this idea, let us build a special set of natural numbers. This set will provide the contradiction we seek. Let D be the set of all non-selfish natural numbers. By definition, the power set P(N) contains all sets of natural numbers, and so it contains this set D as an element. Therefore, D must be paired off with some natural number, say d. However, this causes a problem. If d is in D, then d is selfish because it is in the corresponding set. If d is selfish, then d cannot be a member of D, since D was defined to contain only non-selfish numbers. But if d is not a member of D, then d is non-selfish and must be contained in D, again by the definition of D.

This is a contradiction because the natural number d must be either in D or not in D, but neither of the two cases is possible. Therefore, there is no natural number which can be paired with D, and we have contradicted our original supposition, that there is a bijection between N and P(N).

Note that the set D may be empty. This would mean that every natural number x maps to a set of natural numbers that contains x. Then, every number maps to a nonempty set and no number maps to the empty set. But the empty set is a member of P(N), so the mapping still does not cover P(N).

Through this proof by contradiction we have proven that the cardinality of N and P(N) cannot be equal. We also know that the cardinality of P(N) cannot be less than the cardinality of N because P(N) contains all singletons, by definition, and these singletons form a "copy" of N inside of P(N). Therefore, only one possibility remains, and that is the cardinality of P(N) is strictly greater than the cardinality of N, proving Cantor's theorem.

History[link]

Cantor gave essentially this proof in a paper published in 1891 Über eine elementare Frage der Mannigfaltigkeitslehre, where the diagonal argument for the uncountability of the reals also first appears (he had earlier proved the uncountability of the reals by other methods). The version of this argument he gave in that paper was phrased in terms of indicator functions on a set rather than subsets of a set. He showed that if f is a function defined on X whose values are 2-valued functions on X, then the 2-valued function G(x) = 1 − f(x)(x) is not in the range of f.

Russell has a very similar proof in Principles of Mathematics (1903, section 348), where he shows that that there are more propositional functions than objects. "For suppose a correlation of all objects and some propositional functions to have been affected, and let phi-x be the correlate of x. Then "not-phi-x(x)," i.e. "phi-x does not hold of x" is a propositional function not contained in this correlation; for it is true or false of x according as phi-x is false or true of x, and therefore it differs from phi-x for every value of x." He attributes the idea behind the proof to Cantor.

Ernst Zermelo has a theorem (which he calls "Cantor's Theorem") that is identical to the form above in the paper that became the foundation of modern set theory ("Untersuchungen über die Grundlagen der Mengenlehre I"), published in 1908. See Zermelo set theory.

For one consequence of Cantor's theorem, see beth numbers.

See also[link]

• Controversy over Cantor's theory
• Cantor–Bernstein–Schroeder theorem
• Cantor's first uncountability proof

References[link]

• Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.

Kurt Gödel
Born	(1906-04-28)April 28, 1906 Brno, Moravia, Austria–Hungary
Died	January 14, 1978(1978-01-14) (aged 71) Princeton, New Jersey, United States
Fields	Mathematics, Mathematical logic
Institutions	Institute for Advanced Study
Alma mater	University of Vienna
Doctoral advisor	Hans Hahn
Known for	Gödel's incompleteness theorems, Gödel's completeness theorem, the consistency of the Continuum hypothesis with ZFC, Gödel metric, Gödel's ontological proof
Notable awards	Albert Einstein Award (1951); National Medal of Science (USA) in Mathematical, Statistical, and Computational Sciences (1974) Fellow of the Royal Society[1]
Signature

Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German pronunciation: [ˈkʊʁt ˈɡøːdəl] ( listen); April 28, 1906 – January 14, 1978) was an Austrian/American 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.^[2]

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, 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 (for example 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 neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for "working mathematicians" to assume the axiom of choice in their proofs. There are many important results whose only known proofs use the axiom of choice, which is equivalent to Zorn's lemma. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Life[link]

Childhood[link]

Gödel was born April 28, 1906, in Brno, Austria-Hungary into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh).^[3] At the time of his birth the town had a slight German-speaking majority,^[4] and this was the language of his parents.^[5] The ancestors of Kurt Gödel were often active in the cultural life of the Brno city. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".^[6]

Although he spoke very little Czech himself, Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian empire broke up at the end of World War I. According to his classmate Klepetař, "Gödel considered himself always Austrian and an exile in Czechoslovakia" ("ein Österreicher im Exil in der Tschechoslowakei") during this time.^[7] He chose to become an Austrian citizen at age 23. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.

In his family, young Kurt was known as Herr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.

Gödel attended the Evangelische Volksschule, a Lutheran school in Brno from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

Studying in Vienna[link]

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics.^[8] Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."^[9]

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?

This was the topic chosen by Gödel for his doctorate work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

The Incompleteness Theorem[link]

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 produce a method to encode statements, proofs, and the concept of provability as 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).

The mid 1930s: further work and visits to the US[link]

Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a Privatdozent (unpaid lecturer) there. In 1933 Adolf Hitler came to power in Germany and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, Moritz Schlick, whose seminar had aroused Gödel's interest in logic, was assassinated by a pro-Nazi student. This triggered "a severe nervous crisis" in Gödel.^[11] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.^[12]

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.^[13] He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he delivered a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures which have been subsequently published.

Gödel would visit the IAS again in the autumn of 1935. The traveling and the hard work had exhausted him, and the next year he took a break to recover from a depressive episode. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.

He married Adele Nimbursky (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he.

Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine, in the Blue Hill Inn at the top of the bay. Gödel was taking a vacation from the IAS.

Gödel was not merely vacationing, and had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

Relocation to Princeton, Einstein and US citizenship[link]

After the Anschluss in 1938, Austria had become a part of Nazi Germany. Germany abolished the title of Privatdozent, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down. His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the trans-Siberian railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the U.S. by train to Princeton, where Gödel would accept a position at the Institute for Advanced Study (IAS).

Gödel very quickly resumed his mathematical work. In 1940, he published his work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory which is a classic of modern mathematics. In that work he introduced the constructible universe, a model of set theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo–Fraenkel axioms for set theory (ZF). Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.

Albert Einstein was also living at Princeton during this time. Gödel and Einstein subsequently developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to the Institute merely...to have the privilege of walking home with Gödel".^[14]

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.^[15][16]

Later years and death[link]

Gödel became a permanent member of the Institute of Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.

In 1951, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. He gave this elaboration to Einstein as a present for his 70th birthday.^[17] These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.

During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed.^[18] To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof of God's existence. This is now known as Gödel's ontological proof. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, Adele was hospitalized for six months and could no longer prepare Gödel's food. In her absence, he refused to eat, eventually starving to death.^[19] He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.^[20]

Gravestone of Kurt Gödel in the Princeton, NJ, cemetery

Religious views[link]

Gödel was a convinced theist. He rejected the notion of others like his friend Albert Einstein that God was impersonal.

He believed firmly in an afterlife, stating: "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world [Welt] is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."^[21]

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza."^[22] He said about Islam: "I like Islam. It is a consistent [or consequential] idea of religion and open-minded".^{[23] [24]}

Legacy[link]

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association of Symbolic Logic has invited an annual Kurt Gödel lecture each year since 1990.

Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's Nachlass, and the final two include correspondence.

A biography of Gödel was published by John Dawson in 2005. Gödel was also one of four mathematicians examined in the 2008 BBC documentary entitled "Dangerous Knowledge".^[25]

Douglas Hofstadter wrote a popular book in 1979 called Gödel, Escher, Bach: An Eternal Golden Braid to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain.

Important publications[link]

In German:

• 1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." Monatshefte für Mathematik und Physik 37: 349–60.
• 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I." Monatshefte für Mathematik und Physik 38: 173–98.
• 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger Akademie der Wissenschaften Wien 69: 65–66.

In English:

• 1940. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press.
• 1947. "What is Cantor's continuum problem?" The American Mathematical Monthly 54: 515–25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of Mathematics: Selected Readings. Cambridge Univ. Press: 470–85.
• 1950, "Rotating Universes in General Relativity Theory." Proceedings of the international Congress of Mathematicians in Cambridge, 1: 175–81

In English translation:

• Kurt Godel, 1992. On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
• Kurt Godel, 2000.^[26] On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. Martin Hirzel
• Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press.
  • 1930. "The completeness of the axioms of the functional calculus of logic," 582–91.
  • 1930. "Some metamathematical results on completeness and consistency," 595–96. Abstract to (1931).
  • 1931. "On formally undecidable propositions of Principia Mathematica and related systems," 596–616.
  • 1931a. "On completeness and consistency," 616–17.
• "My philosophical viewpoint", c. 1960, unpublished.
• "The modern development of the foundations of mathematics in the light of philosophy", 1961, unpublished.

• Collected Works: Oxford University Press: New York. Editor-in-chief: Solomon Feferman.
  • Volume I: Publications 1929–1936 ISBN 978-0-19-503964-1 / Paperback:ISBN 978-0-19-514720-9,
  • Volume II: Publications 1938–1974 ISBN 978-0-19-503972-6 / Paperback:ISBN 978-0-19-514721-6,
  • Volume III: Unpublished Essays and Lectures ISBN 978-0-19-507255-6 / Paperback:ISBN 978-0-19-514722-3,
  • Volume IV: Correspondence, A–G ISBN 978-0-19-850073-5,
  • Volume V: Correspondence, H–Z ISBN 978-0-19-850075-9.

See also[link]

	Biography portal
	Logic portal

Notes[link]

References[link]

• Dawson, John W., 1997. Logical dilemmas: The life and work of Kurt Gödel. Wellesley MA: A K Peters.
• 1911 Encyclopædia Britannica/Brünn. (2007, September 19). In Wikisource, The Free Library. Retrieved 10 pm EST March 13, 2008.
• Rebecca Goldstein, 2005. Incompleteness: The Proof and Paradox of Kurt Gödel. W. W. Norton & Company, New York. ISBN 0-393-32760-4 pbk.

Further reading[link]

• John L. Casti and Werner DePauli, 2000. Gödel: A Life of Logic, Basic Books (Perseus Books Group), Cambridge, MA. ISBN 0-7382-0518-4.
• John W. Dawson, Jr. Logical Dilemmas: The Life and Work of Kurt Gödel. AK Peters, Ltd., 1996.
• John W. Dawson, Jr, 1999. "Gödel and the Limits of Logic", Scientific American, vol. 280 num. 6, pp. 76–81
• Torkel Franzén, 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters.
• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
• Jaakko Hintikka, 2000. On Gödel. Wadsworth.
• Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage.
• Stephen Kleene, 1967. Mathematical Logic. Dover paperback reprint ca. 2001.
• Stephen Kleene, 1980. Introduction to Metamathematics. North Holland ISBN 0-7204-2103-9 (Ishi Press paperback. 2009. ISBN 978-0-923891-57-2)
• J.R. Lucas, 1970. The Freedom of the Will. Clarendon Press, Oxford.
• Ernest Nagel and Newman, James R., 1958. Gödel's Proof. New York Univ. Press.
• Procházka, Jiří, 2006, 2006, 2008, 2008, 2O1O. Kurt Gödel: 1906–1978: Genealogie. ITEM, Brno. Volume I. Brno 2006, ISBN 80-902297-9-4. In Ger., Engl. Volume II. Brno 2006, ISBN 80-903476-0-6. In Germ., Engl. Volume III. Brno 2008, ISBN 80-903476-4-9. In Germ., Engl. Volume IV. Brno, Princeton 2008, ISBN 978-80-903476-5-6. In Germ., Engl.Volume V,Brno,Princeton 2O1O, ISBN 8O-9O3476-9-X.In Germ.,Engl.
• Ed Regis, 1987. Who Got Einstein's Office? Addison-Wesley Publishing Company, Inc.
• Raymond Smullyan, 1992. Godel's Incompleteness Theorems. Oxford University Press.
• Olga Taussky-Todd, 1983. Remembrances of Kurt Gödel. Engineering & Science, Winter 1988.
• Hao Wang, 1987. Reflections on Kurt Gödel. MIT Press.
• Hao Wang, 1996. A Logical Journey: From Godel to Philosophy. MIT Press.
• Yourgrau, Palle, 1999. Gödel Meets Einstein: Time Travel in the Gödel Universe. Chicago: Open Court.
• Yourgrau, Palle, 2004. A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books. Book review by John Stachel in the Notices of the American Mathematical Society (54 (7), p 861–868):

External links[link]

Wikimedia Commons has media related to: Kurt Gödel

• O'Connor, John J.; Robertson, Edmund F., "Kurt Gödel", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Godel.html .
• Kurt Gödel at the Mathematics Genealogy Project
• Weisstein, Eric W., Gödel, Kurt (1906–1978) from ScienceWorld.
• Kennedy, Juliette. "Kurt Gödel." In Stanford Encyclopedia of Philosophy.
• Time Bandits: an article about the relationship between Gödel and Einstein by Jim Holt
• "Gödel and the limits of logic" by John W Dawson Jr. (June 2006)
• Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue
• Paul Davies and Freeman Dyson discuss Kurt Godel
• "Gödel and the Nature of Mathematical Truth" Edge: A Talk with Rebecca Goldstein on Kurt Gödel.
• It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities.
• Dangerous Knowledge Google Video of a BBC documentary made by David Malone about the life and work of Kurt Gödel and other revolutionary mathematical thinkers.
• Gödel photo g.
• Kurt Gödel at Find a Grave

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The Foundations were a British soul band, active from 1967 to 1970. The group, made up of West Indians, White British, and a Sri Lankan, are best known for their two biggest hits, "Baby Now That I've Found You" (a Number One hit in the UK Singles Chart and Canada, and subsequently Top 10 in the US), written by Tony Macaulay and John MacLeod; and "Build Me Up Buttercup" (a number 3 on the Billboard Hot 100 and number 1 in Canada), co-written by Macaulay with Mike d'Abo, at the time the lead vocalist with Manfred Mann. The group was the first multi-racial group to have a number 1 hit in the UK in the 1960s.^[1]

The Foundations are notable for being one of the few label acts to successfully imitate what became known as the Motown Sound. In terms of line-up and musical style, they anticipated the sound of the more successful Hot Chocolate. They were in a similar musical vein as Love Affair, who also topped the UK charts in 1968 with their version of Robert Knight's "Everlasting Love". The Foundations signed to Pye, at the time one of only four big UK record companies (the others being EMI with its HMV, Columbia Records, and Parlophone labels; Decca; and Philips who also owned Fontana).^[2]

Biography[link]

Origins[link]

The Foundations drew much interest and intrigue due to the size and structure of the group. Not only was there a diverse ethnic mix in the group, but there was also diversity in ages and musical backgrounds. The oldest member of the group was Mike Elliott, who was 38 years old. The youngest was Tim Harris, who, at 18, was barely out of school. The West Indian horn section, which consisted of Jamaican-born Mike Elliott and Pat Burke, both saxophonists and Dominican-born Eric Allendale on trombone. They were all highly experienced musicians who came from professional jazz and rock-and-roll backgrounds. Mike Elliott had played in various jazz and rock and roll bands including Tubby Hayes and Ronnie Scott,^[3] the Cabin Boys (led by Tommy Steele's brother, Colin Hicks), and others. Pat Burke, a professional musician, was from the London Music Conservatorium. Eric Allendale had led his own band at one stage as well as having played with Edmundo Ros and being a former member of the Terry Lightfoot^[4] and Alex Walsh bands. Alan Warner, the guitarist, was also an experienced musician for his age, having played with numerous semi-pro groups from the age of 16.^[5] Bassist Peter Macbeth was a former teacher. Tony Gomez, the keyboard player, was a former clerk, while Clem Curtis had been an interior decorator and professional boxer.

The story of the origins of the Foundations can be somewhat surprising and a bit confusing as to who was responsible for choosing the band's name, and various sources give slightly different accounts of their beginnings. One version is that they were originally called The Ramong Sound,^[5][6] or The Ramongs, and there were two lead singers, Clem Curtis and Psychedelic shock rocker Arthur Brown.^[7] Other variations on this are that after they dropped the "Ramong" from their name, they were then called The Foundation Squad or The Foundation Sound.^[5] Apparently, Arthur Brown was only a temporary member for about one month, and by the time the Foundations had signed to Pye, he had left the group.

The Foundations actually did come together in Bayswater, London in January 1967. They practiced and played in a basement club called the Butterfly Club, which they ran.^[8] While managing the club themselves, they played music nightly, and handled the cooking and cleaning. They would get to bed around 6 or 7 a.m., sleep until 4 p.m., get up and begin again to get ready to open at 8 p.m. Sometimes they barely made enough money to pay the rent, let alone feed themselves. At times, they lived off the leftovers and a couple of pounds of rice.^[9] They stayed there until they were eventually forced out by a protection racket gang and then had to move next door to a dingy unused mini-cab office.^[10]

Career from 1967[link]

While the biography on Allmusic states that Barry Class was the first to discover them,^[1] Alan Warner, the original founding member and lead guitarist of The Foundations and former Ramongs member states that it was actually Fairway who saw them in the dingy office and then introduced them to Class.^[10] Either way, both Fairway and Class teamed up to become their managers.^[11] Class took care of the business management, while Fairway went out promoting and trying to secure gigs for the group.^[10]

When they were at the top spot with "Baby, Now That I've Found You", Fairway commented to Melody Maker that most management would pull them out of the "bargain priced dates" that were booked for some time. He expressed gratitude to everyone for their support, and said that they would fulfill every engagement for which they had signed.^[3]

Not long after "Baby, Now That I've Found You" became a hit, rock historian Roger Dopson describes what followed as a "behind the scenes struggle",^[3] where Fairway was "pushed out" and his partner, Barry Class, remained as sole manager of the group. Fairway later attempted to sue the band, alleging that he was wrongfully dismissed, though the band said that he had resigned of his own accord.^[12] Dopson also noted that Fairway also leaked a story to the media saying that the Foundations had broken up which only served to keep the Foundations name in the news headlines.^{[3] [13]} Fairway and Class eventually introduced them to Pye A&R man Tony Macaulay.^[10] The day Macaulay came to hear them play, he was suffering from what he described as the worst hangover of his life. The band was playing so loud he could not judge how good they were, but he decided to give them a chance.^[3] He would later comment in the book, 1000 UK #1 Hits by Jon Kutner and Spencer Leigh, that he woke up that morning with a stinking headache, and when he got to the studio and heard The Foundations, he thought they were pretty terrible. He decided his hangover was to blame, and so he gave them the benefit of the doubt.^[14]

At first they found progress quite slow, and one of their sax players, Pat Burke, had to drop out of the band and take another job while they went through a rough patch. He did rejoin them again later in 1967.^{[15] [16]}

They did a couple of tours backing The Toys and later Motown Records' Edwin Starr.^[10] Curtis doubted if this group called The Toys was the original Toys let alone American. They were noticed by Brian Epstein who added them to the roster of his NEMS Agency but the contract became void when he died.^[17]

When "Baby Now That I've Found You" was first released it went nowhere. Luckily the BBC's newly founded BBC Radio 1 were looking to avoid any records being played by the pirate radio stations and they looked back at some recent releases that the pirate stations had missed. "Baby, Now That I've Found You" was one of them. The single then took off and by November was number one in the UK Singles Chart.^[2] This was the ideal time because of the soul boom that was happening in England since 1965 and with American R&B stars visiting the UK, interest and intrigue in The Foundations was generated. Their second single released in January 1968, "Back On My Feet Again", did not do as well but made it to #18 in the UK,^[2] and #29 in Canada. Also in January 1968 they were invited to put down some tracks for John Peel's radio show. One of the tracks that they laid down was a cover of ? and the Mysterians garage classic 96 Tears.^[18][19] On the same day PP Arnold was in the studio with Dusty Springfield and Madeline Bell as her backing vocalists.^[18]

Around this time after the release of their second single, there were tensions developing between the band and their songwriter/producer, Tony Macaulay. He would not allow them to record any of their own songs.^[1] In an interview, the band's organ player, Tony Gomez, told NME Magazine in an interview that he, Peter MacBeth, and Eric Allendale had some ideas that they wanted to put together. Curtis later recalled that Macaulay was a problem. "Tony Macaulay was very talented, but could be difficult to get on with. When we asked to record some of our own material - just as B sides, we weren't after the A side - he called us 'ungrateful' and stormed out of the studio."^[3] The group felt that Macaulay had reined in their "real" sound, making them seem more pop-oriented than they were.^[1] Tony Macaulay was later to recall, "I was never close to The Foundations. I couldn't stand them, and they hated me! But the body of work we recorded was excellent."^[3]

A third single, also released in 1968 "Any Old Time (You're Lonely and Sad)", reached #48.^[2]

Curtis and Elliott leave group[link]

Original vocalist Curtis left in 1968, because he felt that a couple of the band's members were taking it a bit too easy, thinking that because they had now had a hit, they did not have to put in as much effort as they had previously.^[1] Saxophonist Mike Elliott also left around this time and was never replaced. Curtis hung around and helped them audition a replacement singer. They auditioned 200 singers. It was reported in a New Musical Express article in 1968 that Curtis while being interviewd at a festival had mentioned that they were trying out Warren Davis to replace him. He said he wouldn't leave the band until they found a replacement.^[20] He had become friendly with Sammy Davis, Jr. and was encouraged to try his luck in the United States. He moved to the US for a solo career on the club circuit, encouraged by the likes of Wilson Pickett and Sam & Dave, playing Las Vegas with the Righteous Brothers. His successful replacement was Colin Young.

New lead singer[link]

With Young the band had two more big hits; "Build Me Up Buttercup" which was their third hit in 1968 and "In the Bad Bad Old Days (Before You Loved Me)" which was a hit in April 1969,^[2] and reached #23 in Canada May 5 that year.

At the height of their popularity, the Foundations management were in negotiations and a UK TV company for a television series that would star members of the band. They had turned down a number of offers to appear in films because of script unsuitability.^[21]

Bassist Peter Macbeth left the band in 1969, to join the group Bubastis with Bernie Living,^[22][23] and was replaced by Steve Bingham.

Beginning of 1970 to the breakup in late 1970[link]

After a successful run of hits, The Foundations broke off with their management and a Bill Graham sponsored tour to support The Temptations at the newly opened Copacabana club. This ended up in disaster and the band came back to the UK in low spirits. It had been previously reported in a publicity sheet around early December 1969 that the band had broken away from their manager Barry Class. Jim Dawson who was formerly their agent and Mike Dolan took over the group's affairs.^[24] The group's final hits were "Born to Live, Born to Die" which was written by Eric Allendale and Tony Gomez.^[25] and "My Little Chickadee", a US only hit which barely made the hot 100. Another member joined the band in 1970. Paul Lockey who had been with Robert Plant in Band Of Joy joined as their bass guitarist.^[26]

"My Little Chickadee" proved to be the band's last hit. In spite of releasing "Take A Girl Like You", the title song to the Oliver Reed and Hayley Mills film, and a heavy blues rock song "I'm Gonna Be A Rich Man", the band split in late 1970.^[27]

1971 to the end of the 1970s[link]

The last record released in the early 1970s as The Foundations was a single "Stoney Ground" b/w "I'll Give You Love" MCA MCA 5075 1971.^[28] By that time the original band had already split up and Alan Warner had been a member of Pluto since joining and helping to form the English progressive rock band in 1970.^[10] There would be two more singles released as The Foundations in the mid to late 1970s.

When Curtis returned to the UK, he formed a new version of the group with little success in spite of releasing several singles, but later had a lucrative spell on the 1960s nostalgia circuit. Curtis' re-formed Foundations have on several occasions and among the many musicians to be part of latter day Foundations were Bill ^[29] and John Springate, the latter becoming a member of The Glitter Band,^[30] Derek "Del" Watson, Paul Wilmot (all members of the band "ELEGY") and Roy Carter who later on joined Heatwave.^[31]

Also in the 1970s there would be a collaborative attempt between two former members of the Foundations. Original Foundations trombonist Eric Allendale attempted to work with original Foundations Foundations drummer Tim Harris.^[32]

In the mid 1970s, while Clem Curtis and The Foundations were on the road, there was also another Foundations line up that was led by Colin Young who were on the road at the same time, who were playing basically the same material. This eventually led to court action which resulted in Curtis being allowed to bill his group as either The Foundations or Clem Curtis & The Foundations. Young was allowed to bill himself as The New Foundations or Colin Young & The New Foundations.^[3]

Also in the mid 1970s, Young and his group, The New Foundations, released a lone single on Pye, "Something For My Baby" / "I Need Your Love".^[3] There were actually two more singles released in the late 1970s as the Foundations. They were "Where Were You When I Needed Your Love" / "Love Me Nice And Easy" and "Closer To Loving You" / "Change My Life" on the Summit and Psycho labels.^[33] These featured Curtis as the lead singer.

Various sources erroneously state that there was an early 1970s English line up that had nothing, or little to do with, the original Foundations. However, Curtis has been leading a new line up of the Foundations since coming back to the UK and reforming the group in the early 1970s.

1980s to present[link]

In or around the late 1980s Curtis and Alan Warner teamed up to recut "Baby, Now That I've Found You" as well as other hits of The Foundations.^[3]

There has also been another line up formed in 1999 that included Young (vocals), Alan Warner (Guitar), Steve Bingham (bass), Gary Moberly (keyboards), Tony Laidlaw (sax) and Sam Kelly then Steve Dixon (drums). This version of the group was reformed due to the popularity of the film There's Something About Mary and the interest created resulting from the 1968 hit "Build Me Up Buttercup" being featured in the film. Some time later Young left this version of the group and was replaced by Hue Montgomery (aka Hugh Montgomery).

Known as Alan Warner's Foundations, this group is on tour in 2012 either in their own right or with The New Honeycombs, Terry Rice-Milton of Cupid's Inspiration, Billie Davis and Keith Powell.

Curtis still appears at venues and tours as Clem Curtis & The Foundations as he has done during the past four decades.

There is a bogus American group using the Foundations name and includes original Foundations hits as part of their repertoire. This group has no connection with the group that recorded the hits on Pye.^{[citation needed]}

Former personnel[link]

The Foundations[link]

• Clem Curtis: lead vocals - born 28 November 1940,^[34] Trinidad, West Indies.
• Colin Young: lead vocals - b. 12 September 1944, Barbados, West Indies - replaced Clem Curtis in 1968.
• Arthur Brown: vocals - b. 24 June 1942 Whitby, Yorkshire, Member for approximately one month in 1967
• Alan Warner: lead guitar - b. 21 April 1947,^[34] Paddington, west London.
• Peter Macbeth: bass guitar - b. Peter McGrath, 2 February 1941,^[34] Marylebone, North London.
• Steve Bingham: bass guitar - b. 4 April 1949, Solihull, Warwickshire.
• Tim Harris: drums - b. 14 January 1948,^[34] St John's Wood, North London - Died 2007
• Tony Gomez: keyboards - b. 13 December 1938,^[34] Colombo, Ceylon - (now Sri Lanka).
• Pat Burke: tenor saxophone/flute - b. 9 October 1937,^[34] Kingston, Jamaica, West Indies.
• Mike Elliott: tenor saxophone - b. 6 August 1929,^[34] Jamaica, West Indies. - Left in 1968
• Eric Allandale: trombone - b. Eric Allandale Dubuisson, 4 March 1936,^[34] Dominica, West Indies — died 23 August 2001.
• Paul Lockey: bass guitar^[35] - joined in 1970 for nine months

Guests[link]

• Mike D'Abo :piano - b. Michael David D'Abo, 1 March 1944, Betchworth, Surrey. Co-wrote and guested on "Build Me Up Buttercup" contributing piano.
• John Mcleod: piano

Clem Curtis & The Foundations[link]

• Clem Curtis: vocals
• James Colah: keyboards
• Michael J. Parlett: saxophone
• Alan Warner: guitar ^[3]
• Roy Carter: bass guitar ^[36]
• George Chandler: backing vocals
• Valentine Pascal: electric guitar

1970s line-up[link]

• Clem Curtis: Vocals
• Bill Springate: guitar
• John Springate: Vocals
• Del Watson: Bass
• Paul Wilmot : Drums
• John Paul: ? ^[37]

1977 line-up[link]

• Clem Curtis: Vocals
• Leroy Carter
• John Savile
• Valentine Pascal
• Georges Delanbanque ^[38]

Discography[link]

Summary of single releases

From the bands beginning to their breakup near the end of 1970, the Foundations released ten singles in the UK including two versions of the same song. A good deal of the songs on the singles were composed by Tony Macaulay and John Macleod. They had four significant hits from these plus a minor hit with one of their own compositions, "Born To Live, Born To Die" and another minor hit with "My Little Chickadee" in the United States. There were other titles announced that were either never recorded or were never released. They were "Our Love Went Thataway",^[39] "Tear Jerker, Music-worker, You" which was to be released around the same time as "Better By Far" by Lulu and "No Place On Earth Could Find You" ^[40][41] In 1971 the single "Stoney Ground" was released. It is believed that this single was actually by Colin Young and his new backing band Development. It seems quite likely as the Colin Young and Development debut single "Any Time At All" pre-dates "Stoney Ground". In the mid and late seventies there were two more singles released under the Foundations name. They were "Where Were You When I Needed Your Love" and "Closer To Loving You" which featured the Northern Soul classic "Change My Life" as the B side. These last two singles to bear the Foundations name featured Clem Curtis once more as the lead vocalist.

Summary of album releases

During the 1960s the Foundations recorded and released four LPs in the United Kingdom. Before the release of their debut album, it was originally announced, in the October 1967 of Beat Instrumental Monthly that the debut album's title was to be Sound Basis.^[9] However, when it was released on Pye, it had the title of From The Foundations. The American version of this album, which was released on the UNI label, was given the title of Baby, Now That I've Found You. This album featured Curtis on lead vocals. The next release was in 1968. It was a live LP called Rocking The Foundations, and also featured Curtis on lead vocals plus two instrumentals "The Look of Love" and "Coming Home Baby". Also in 1968, another LP was released on the Marble Arch label. This featured re-recordings of their previous hits and songs, but with Young on vocals instead of Curtis. It also featured a version of "Build Me Up Buttercup". There was also an American album called Build Me Up Buttercup. This was a compilation of Foundations tracks. Side one consisted of tracks from their Rocking The Foundations album; and side two consisted of "Build Me Up Buttercup", the B side of that single, plus some earlier Foundations tracks. The group's last LP release was in 1969, which featured their hit "In The Bad Bad Old Days", and the minor US hit "My Little Chickadee". Since then there have been various compilations of the Foundations songs, released on both the Golden Hour and PRT labels.

UK singles[link]

• "Baby, Now That I've Found You" / "Come On Back to Me" - Pye 7N 17366 - 1967 - UK #1, US #11
• "Back On My Feet Again" / "I Can Take Or Leave Your Loving" - Pye 7N 17417 - 1968 - UK #18
• "Any Old Time (You're Lonely And Sad)" / "We Are Happy People" - Pye 7N 17503 -1968 - UK #48
• "Build Me Up Buttercup" / "New Direction" Pye 7N 17636 - 1968 - UK #2, US #3
• "In the Bad Bad Old Days (Before You Loved Me)" / "Give Me Love" - Pye 7N 17702 - 1969 - UK #8
• "Born to Live, Born to Die" / "Why Did You Cry" - Pye 7N 17809 1969 - UK #46
• "Baby, I Couldn't See" / "Penny Sir" - Pye 7N 17849 - 1969
• "Take A Girl Like You" / "I'm Gonna Be A Rich Man" - Pye - 7N 17904 - 1970
• "I'm Gonna be a Rich Man" / "In The Beginning" Pye 7N 17956 - 1970
• "Stoney Ground" / "I'll Give You Love" - MCA MKS 5075–1971
• "Where Were You When I Needed Your Love" / "Love Me Nice And Easy" - Summit SU 100
• "Closer To Loving You" / "Change My Life" - Psycho P 2603–1978
• "Baby Now That I've Found You" / "Build Me Up Buttercup" - Old Gold OG9407 - (1979 Re-release)
• "Baby Now That I've Found You" / "Build Me Up Buttercup" - Flashback FBS 6 - (1979 Re-release)
• "Baby Now That I've Found You" (1989 Re Mix) / "Build Me Up Buttercup" - PRT PYS 24 - 1989
• "Build Me Up Buttercup" - CD single - (1998 Re-release) - UK #71

^{[2] [33]}

UK albums[link]

• From the Foundations - Pye NSPL 18206 - 1967
• Rocking the Foundations - Pye NSPL 18227 - 1968
• The Foundations - Marble Arch MALS 1157–1968
• Digging the Foundations - Pye NSPL 18290 - 1969
• Golden Hour of the Foundations (Greatest Hits) - GH 574 - 1973
• Back to the Beat - PRT DOW7 - 1983
• Best Of - PRT PYL 4003–1987

UK EPs 7"[link]

• "It's All Right" - Pye NEP24297 - 1968
• "Mini Monster" - Pye PMM.103

UK EPs 12"[link]

• "Baby, Now That I've Found You" - Pye Big Deal BD 107 - (4 tracks)
• "Baby, Now That I've Found You" - PRT Pyt 24 - 1989 - (3 tracks incl remix)

UK CDs[link]

• Golden Hour Of The Foundations - Knight Records KGH CD 104 - 1990
• Strong Foundations - The Singles and More - Music Club - MCCD 327 - 1997
• Build Me Up Buttercup - Castle Select SELCD 527 - 1998
• Baby, Now That I've Found You - Sequel Records - NEECD 300 - 1999
• Build Me Up Buttercup (The Complete Pye Collection) [Remastered] - Castle - 2004