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Cantor's diagonal argument

In set theory, Cantor's diagonal argument called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began; the diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are also the source of contradictions like Russell's paradox and Richard's paradox. In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits, he begins with a constructive proof of the following theorem: If s1, s2, …, sn, … is any enumeration of elements from T there is always an element s of T which corresponds to no sn in the enumeration.

The proof starts with an enumeration of elements from T, for example: Next, a sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s1, the 2nd digit as complementary to the 2nd digit of s2, the 3rd digit as complementary to the 3rd digit of s3, for every n, the nth digit as complementary to the nth digit of sn. For the example above, this yields: By construction, s differs from each sn, since their nth digits differ. Hence, s cannot occur in the enumeration. Based on this theorem, Cantor uses a proof by contradiction to show that: The set T is uncountable; the proof starts by assuming. All its elements can be written as an enumeration s1, s2... sn.... Applying the previous theorem to this enumeration produces a sequence s not belonging to the enumeration. However, this contradicts s therefore belonging to the enumeration; this contradiction implies. Therefore, T is uncountable; the interpretation of Cantor's result will depend upon one's view of mathematics. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T.

It does not rule out the possibility. In the context of classical mathematics, this is impossible, the diagonal argument establishes that, although both sets are infinite, there are more infinite sequences of ones and zeros than there are natural numbers; the uncountability of the real numbers was established by Cantor's first uncountability proof, but it follows from the above result. To prove this, an injection will be constructed from the set T of infinite binary strings to the set R of real numbers. Since T is uncountable, the image of this function, a subset of R, is uncountable. Therefore, R is uncountable. By using a method of construction devised by Cantor, a bijection will be constructed between T and R. Therefore, T and R have the same cardinality, called the "cardinality of the continuum" and is denoted by c or 2 ℵ 0. An injection from T to R is given by mapping strings in T to decimals, such as mapping t = 0111... to the decimal 0.0111.... This function, defined by f = 0.t, is an injection because it maps different strings to different numbers.

Constructing a bijection between T and R is more complicated. Instead of mapping 0111... to the decimal 0.0111... it can be mapped to the base b number: 0.0111...b. This leads to the family of functions: fb = 0.tb. The functions f b are injections, except for f 2; this function will be modified to produce a bijection between T and R. A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all subsets of S —has a larger cardinality than S itself; this proof proceeds as follows: Let f be any function from S to P. It suffices to prove; that means that some member T of P, i.e. some subset of S, is not in the image of f. As a candidate consider the set: T =. For every s in S, either s is in T or not. If s is in T by definition of T, s is not in f, so T is not equal to f. On the other hand, if s is not in T by definition of T, s is in f, so again T is not equal to f. For a more complete account of this proof, see Cantor's theorem.

This result implies. If S were the set of all sets P would at the same time be bigger than S and a subset of S. Russell's Paradox has shown us that naive set theory, based on an unrestricted comprehension scheme, is contradictory. Note that there is a similarity between the construction of T and the set in Russell's paradox. Therefore, depending on how we modify the axiom scheme of comprehension in order to avoid Russell's paradox, arguments such as the non-existence of a set of all sets may or may not remain valid; the diagonal argument shows that the set of real numbers is "bigger" than the set of natural numbers. Therefore, we can ask if there is a set whose cardinality is "between" that of the integers and that of the reals; this question leads to the famous continuum hypothesis. The question of whether there exists a set whose cardinality is between

Jordan Gibson

Jordan Lewis Gibson is an English professional footballer who plays for Bradford City, as a winger. Born in Birmingham, Gibson played youth football with Rangers, joining the Scottish club in June 2015. After being released, he went on trial with Bradford City, signing a one-year contract with the club in August 2017, he made his professional debut on 29 August 2017, in a Football League Trophy game, started to be considered part of the first team squad in November 2017. The next month he stated, he scored his first goal for Bradford in a 2-1 loss against Oldham Athletic on 3 February 2018. In May 2018 he signed a new two-year contract with Bradford City; that month he criticised comments made about him by former City manager Simon Grayson. In September 2018 he stated that he wished to play more first-team football, praised young teammates Eliot Goldthorp and Reece Staunton. In January 2019 he was linked with a loan move to Stevenage; the deal was completed on transfer deadline day, 31 January 2019.

After his loan deal ended and he returned to Bradford City, new manager Gary Bowyer confirmed that he was in his first-team plans for the 2019–20 season. After starting the first game of the season, Gibson said that his self-belief had been encouraged by Bowyer. Gibson missed City's first victory of the season due to a hamstring injury, remained injured into October 2019. By early December he was nearing fitness; as of 2 February 2020

Lypetska Polyana

Lipetsk Polyana is a village in Khust Raion of Zakarpattia Oblast, part of the Transcarpathian region of Ukraine. The center of the village council, located on the right bank of the river Rika is 22 km from the district center and the railway station Khust, on the highway Khust - Svalyava; the population at the 2001 census was 2,965 people. The first mention of Lipetsk Polyana dates back to 1653. In March-April 1919, Soviet power existed in the village. In 1932, the organization of the HR Committee began to operate, whose first secretary was I. Perez. After the liberation of the Lipetsk Polyany from the Nazi occupiers, 23 residents voluntarily entered the Red Army, 9 were part of the 1st Czechoslovak Army Corps. 2 of them were awarded with orders and medals of the USSR, 18 gave their lives in battles for freedom and independence of the Motherland. In the village there is a monument to fellow villagers. In 1930, the Jewish population was 239. With the Hungarian occupation of the region in March, 1939, Jews were forcefully removed from their occupations.

From 1940 to 1941, Jews from the village were drafted into forced labor battalions and others were drafted for service on the Eastern front, where most died. By 1941, the Jewish population had increased to 280 and Jewish families without Hungarian citizenship were expelled to Nazi occupied Ukrainian territory, to Kamenets-Podolski, murdered there; the remaining Jews of Lypetska Polyana were deported to Auschwitz late May, 1944. Zakarpattia Oblast

1916 Eastern Suburbs season

The 1916 Eastern Suburbs DRLFC season was the 9th in the club's history. They competed in the 1916 NSWRFL season. Eastern Suburbs won the City Cup for the third consecutive season. Premiership Round 1, Saturday 6 May 1916,Eastern Suburbs 5 defeated Annandale 2 at Wentworth Park. Premiership Round 2, Saturday 13 May 1916,Newtown 11 defeated Eastern Suburbs 5 at the Sydney Cricket Ground. Premiership Round 3, Saturday 20 May 1916,North Sydney 6 defeated Eastern Suburbs 4 at the Sydney Sports Ground. Premiership Round 4, Saturday 27 May 1916,South Sydney 9 defeated Eastern Suburbs 7 at the Agricultural Ground. Premiership Round 5, Saturday 3 June 1916,Eastern Suburbs 12 defeated Western Suburbs 5 at St Luke's Park. Premiership Round 6, Monday 5 June 1916,Eastern Suburbs 2 drew with Glebe 2 at the Agricultural Ground. Premiership Round 7, Saturday 10 June 1916,Balmain 5 defeated Eastern Suburbs 4 at the Sydney Cricket Ground. Premiership Round 8, Saturday 17 June 1916,Eastern Suburbs 13 defeated Annandale6 at Wentworth Park.

Premiership Round 9, Saturday 24 June 1916. Crowd 3, 000. Played at the Sydney Cricket Ground, Following Newton's kick off, Challis found the line cleverly, a moment or two latter Messenger kicked a penalty goal. Eastern Suburbs, 2 point to nil. Newtown's backs combined nicely, but Bain missed a penalty from an easy position. Newtown were driven back in turn by Gilbert and Watkins. Once a fine effort by Ryan, from a breaking scrummage looked dangerous but the subsequent faulty combination of Talbot and Brown spoilt the opportunity. Newtown's lack of finish a few minutes caused a break down just when a score seemed imminent. Rigney retired with an injured knee’ Bain falling back to the last line whilst Collins was requisitioned for the wing. Play followed for a while, Capless after a splendid pass from Male, made a brilliant fending run and scored. Messenger scored a goal. Eastern Suburbs, 7 points to nil. Ryan, the best forward game on the ground, put in a great effort and with support carried play to the Eastern Suburbs twenty-five.

Here Gillespie short kicked over the line, M Cue following fast scored. Bain failed at goal. Eastern Suburbs 7 point-. Newtown reorganised their back division. Bain coming back to centre, Brown acted as fullback, the result proved the efficiency of this alteration. Eastern Suburbs were formidable for a while, only the Alertness of Collins on one occasion saved the situation for Newtown; the Newtown forwards, with Ryan and Townsend prominent, began to trouble their opponents and after a succession of snappy attacks, Bain kicked for the comer, Collins made a good try. Brown's kick dropped short Eastern Suburbs 7 points, Newtown 6. Play became exciting and both teams put forward great efforts, after many tense moments Newtown came again with an irresistible rush and from the scramble Larkin scored, Bain missed the goal. Newtown 9, Eastern Suburbs 7. A few minutes Messenger made scores equal with a neat penalty goal - 9 points each. Again the Newtown pack rushed play to their opponents' zone, Bain again using the cross-kick cleverly, paved the way for Collins to score his second try P. A. McQue converted, Newtown 14 points to 9.

Eastern Suburb’s prospects appeared poor but In a flash an opportunity came, a scrummage at midfield gave Male possession, the half-back sent out a lob pass to Cubltt, who took the ball cleverly and went through the opposition with a characteristic dodgy run. Gilbert joined in, carried the ball to well within Newtown twenty-five, where a final pass sent White in between'the posts. A splendid try, Messenger scored a goal. 14 points all This were the full time score.. Sydney Morning Herald Premiership Round 10, Monday 26 June 1916,Eastern Suburbs 12 defeated North Sydney 6 at Agricultural Ground. Premiership Round 11,??? 1 July 1916,Eastern Suburbs 24 defeated South Sydney??. South, who were unbeaten, led 10-9 with just 15 minutes remaining; the Tricolours finished with two late tries to Les Cubbitt and one to fullback Harry Caples, in what was described as "one of the most exciting finishes to a match in years". Premiership Round 12, Saturday 8 July 1916,Eastern Suburbs 53 ( Gilbert 4 defeated Western Suburbs 0 at the Sydney Sports Ground.

This was the club's biggest win against Wests and remained so until passed by the, Phil Gould, Roosters outfit in 1998. Premiership Round 13, Saturday 15 July 1916'Eastern Suburbs 8 defeated Glebe 5 at Agricultural Ground. Crowd 10,000 Walter Messenger's absence was keenly felt by Eastern Suburbs, as a moment after A Bolewski's kick-of, the emergency, missed two penalties, from comparatively easy ranges. Play was taken to Glebe's territory, but a penalty sent tho ball back L. Burge was conspicuous for Glebe, but his transfer for Frank Burge went astray just as an excellent opening presented itself; the game was good, with both combinations exerting their utmost. The ‘spotting’ of the respective halfbacks and centres was excellent and Watkins for Eastern Suburbs, doing some good work. Glebe, were playing and, although frequcnt

Turn My Back on You

"Turn My Back on You" is a song by English band Sade from their third studio album, Stronger Than Pride. It was released as the album's fourth single on 21 November 1988. Sophie Heawood of The Guardian commented, "Anchored by a bassline that feels like it could go on for Sade's light touch defines this, her casualness and distracted ba-ba-bas belie her devotion, but it's all in the details: the crucial pause in the way she sings'You are my... religion,' for instance." Frank Guan of Vulture wrote, "Complicated lyrics would only get in the way of that massive bass line. Sade sticks to plain professions of fidelity and leaves it to Paul Denman to carry the day." 7-inch singleA. "Turn My Back on You" – 4:11 B. "Keep Looking" – 5:2112-inch singleA. "Turn My Back on You" – 6:09 B1. "Turn My Back on You" – 7:06 B2. "Keep Looking" – 5:21UK and European CD single"Turn My Back on You" – 6:09 "Turn My Back on You" – 7:06 "Keep Looking" – 5:21 Lyrics of this song at MetroLyrics


Herbimycin is a benzoquinone ansamycin antibiotic that binds to Hsp90 and alters its function. Hsp90 client proteins play important roles in the regulation of the cell cycle, cell growth, cell survival, apoptosis and oncogenesis, it was found by its herbicidal activity, thus named. The most recent herbimycins to be discovered, herbimycins D-F, were isolated from a Streptomyces isolated from thermal vents associated with the Ruth Mullins coal fire in Appalachian Kentucky. Antibiotic Tan 420F Herbimycin A Herbimycin induces the degradation of proteins that are need to be mutated in tumor cells such as v-Src, Bcr-Abl and p53 preferentially over their normal cellular counterparts; this effect is mediated via Hsp90. Geldanamycin Satoshi Ōmura Herbimycin A from Center for Pharmaceutical Research and Innovation