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Topic: Diagonal lemma

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	Encyclopedia: List of lemmas
		In mathematical logic, the diagonalization lemma states that for any well formed formula with a free variable x, there is a sentence ψ such that where [ψ] is the Gödel number for ψ.
		Fatous lemma establishes an inequality relating the integral (in the sense of Lebesgue) of the limit inferior of a sequence of functions to the limit inferior of the sequence of integrals of the functions.
		Sards lemma, also known as Sards theorem or the Morse-Sard theorem, is a result of mathematical analysis characterising the image of the critical points of a smooth function F from one Euclidean space to another as having Lebesgue measure 0 (and so small, in a definite sense).
	www.nationmaster.com /encyclopedia/List-of-lemmas (1813 words)

	Diagonal lemma - Wikipedia, the free encyclopedia
		In mathematical logic, Gödel's diagonal lemma is a precise way of constructing self-referential statements.
		Let T be a theory in an extension of the language of arithmetic in which all recursive functions are representable.
		I.e., the result of substituting the quotation name of A for x in A. This mapping is called diagonalization, and D(A) is the diagonalization of A, and D is called the diagonal function.
	www.wikipedia.org /wiki/Diagonal_Lemma (214 words)

	diagonal
		As applied to a polygon, a diagonal is a line segment joining two vertices that are not adjacent.
		When n is the number of vertices in a polygon and d is the number of possible different diagonals, each vertex has possible diagonals to all other vertices save for itself and the two adjacent vertices, or n-3 diagonals; this multiplied by the number of vertices is (n-3).
		In the case of a square matrix, the main or principal diagonal is the diagonal line of entries running north-west to south-east.
	www.fact-library.com /diagonal.html (366 words)

	MAT 200 Lecture Notes -- More About Inverses
		The diagonal line from the upper left hand corner of a square matrix to the lower right hand corner is called the main diagonal.
		We have already seen the main diagonal once before, in the definition of identity matrices; the non-zero entries were the ones on the main diagonal.
		The sum of the entries on the main diagonal of a square matrix a is called the trace of a and is denoted tr(a).
	www.math.princeton.edu /~stalker/200s00/onot.html (549 words)

	Re: Undecidability
		It is only by scrutinizing the proof of the > diagonal lemma and the other steps in the proof that it can be decided > if they offend any philosophical sensibilities, and if so, how.
		Perhaps it's the very "diagonal" look of the statement of the diagonal lemma that makes people suspicious.
		Perhaps it's worthy to remind that the diagonal lemma is not essential in the proof.
	philo.at /phlo/199605/msg00555.html (212 words)

	lemma
		In mathematics, a lemma expresses a minor theorem, of interest primarily because it serves in the proof of a major result.
		In linguistics, and particularly in morphology, a lemma comprises the canonical form of a word.
		Lemmas have especial significance in highly inflected languages such as Czech.\n
	www.fact-library.com /lemma.html (60 words)

	MATH 7200 - Exam 1, question 4 (Site not responding. Last check: 2007-09-19)
		The lemma below states that the diagonals of a square are perpendicular bisectors.
		The last two vertices (points C and D) are found at the intersection of the circle and the perpendicular bisector.
		LEMMA: Given a square ABCD, the diagonals are perpendicular bisectors of each other.
	jwilson.coe.uga.edu /EMT668/EMT668.Folders.F97/Weber/MATH7200/exam1/1test4.html (146 words)

	Top: SelfTestingCorrecting
		In WeakPCPTheorem, we use self testing and correcting of "diagonal" linear functions to show a weak version of the PCP theorem.
		LEMMA: If F is approximately linear, then, for any W∈ A, with probability at least 9/10, SELF-CORRECT(F,W) outputs G(W), where G is the linear function encoded by F. That there is a unique linear function encoded by F follows from the lemmas above.
		DEFN: F is an approximately diagonal linear function if F is approximately linear and the linear function that F encodes is a diagonal linear function.
	www.cs.ucr.edu /~neal/wiki/wiki.pl?SelfTestingCorrecting (854 words)

	[No title]
		Any real symmetric matrix is orthogonally equivalent to a diagonal matrix.\\ \noindent{\bf 11.22 Lemma} If a complex (real) matrix $A$ is unitary (resp., orthogonally) equivalent to a diagonal matrix with real diagonal entries, then $A$ is Hermitean (resp., symmetric).\\ {\em Proof}.
		By 11.21 and 12.10, $A=P^{-1}DP$, where $D$ is a diagonal matrix with positive diagonal entries and $P$ a unitary (orthogonal) matrix.
		Next, the faster the subdiagonal entries $a_{i+1,i}^{(k)}$ converge to zero the faster the diagonal entries $a_{ii}^{(k)}$ converge to the eigenvalues of $A$.\\ One can modify the matrix $A$ to decrease the ratio $\lambda_{n}/\lambda_{n-1}$ and thus accelerate the convergence of $a_{nn}^{(k)}\to\lambda_n$, with the help of shifting, as in 19.12.
	www.math.uab.edu /chernov/teaching/632notes (10997 words)

	Diagonal Lemma - self-reference
		The *diagonalization* of a formula phi with one free variable is the
		(6) the diagonalization of "the diagonalization of x is false" is false.
		Lemma to construct an arithmetical sentence G such that G is equivalent to a
	www.groupsrv.com /science/about13720.html (1266 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		(Please refer to the detailed proof of lemma 4 and lemma 5 in Chapter 1.2.3 of the text book) \vspace{0.1in} \begin{proof} Lemma 3 is proved by induction on the number of vertices $n$.
		If the diagonal does not intersect with any edge of the graph, then an ear is found.
		\end{lemma} %\fig{lct2.eps}{figure_lct2_x}{There must be a diagonal} \begin{figure}[htbp] \begin{center} \psfig{figure=lct2.eps,width=2.0 in} \end{center} \caption{There must be a diagonal} \label{fig:diagonal_exists} \end{figure} Prove that every polygon may be partitioned into triangles by the addition of diagonals.
	web.engr.oregonstate.edu /~saurabh/cs419-519/scribe/lec2/lct.tex (621 words)

	[No title]
		In an appendix independent of the rest of the paper, we use ideas from Goodwillie calculus to show that such natural stable splittings are uniq* *ue, and discuss three different constructions showing their existence.
		DIX: STABLE SPLITTINGS AND THE DIAGONAL 5 The proof of [K4, Lemma 2.6], based on the naturality of the transfer and the behavior of extended power constructions under conjugation, generalizes to the setting here to prove the next lemma.
		X^Ba is a diagonal map into a smash product of more than one copy of X. If X is a coH-space, then will be (nonequivariantly) null, so that ^G=Ga will be G-equivariantly null, establishing the lemma.
	hopf.math.purdue.edu /Kuhn/kuhnsplit.txt (3714 words)

	[No title]
		Definition 4.1.1.Let G 2 GL(n; k) and d be a diagonal of G. A twist along d, denoted by rd(G), is the element of GL(n; k) obtained by `breaking' G along d into two parts, reflecting (or `twisting') one of the pieces, and `gluing' them* * back together (Figure 15).
		In the language of polygons, this means rd does not affect the cel* l, where d is the diagonal* representing the double point p3.
		Definition 4.3.3.A marked twist of an n-gon G along its diagonal d, denoted by erd(G), is the polygon obtained by breaking G along d into two parts, reflectin* *g the piece that does not contain the side labeled 1, and gluing them back together.
	hopf.math.purdue.edu /Devadoss/mosaic.txt (6493 words)

	Online Papers (Site not responding. Last check: 2007-09-19)
		It is often said that diagonalization allows one to construct sentences that are self-referential.
		This sentence is intuitively inconsistent, but the sentence constructed by using diagonalization in the usual way is true and, in fact, provable in Q. This problem can be resolved by expanding the language to include function-symbols for all primitive recursive functions.
		It can also be resolved by proving a stronger form of the diagonal lemma that I call the "structural" diagonal lemma.
	bobjweil.com /heck/philosophy/online_papers.php (1895 words)

	[No title]
		A lemma which I started to state in general, but which I decided to give only the special case in class, verifies that V_i is in fact an L-invariant subspace (try as an exercise): Suppose L, M are commuting linear maps V -> V, that is LM = ML.
		The key lemma for a linear map A: V -> V which lets us show that it can be diagonalized with respect to an orthonormal (unitary) basis when self-adjoint (symmetric or hermitian) is this: if a subspace W is A-invariant, then its orthogonal complement W^\perp is A*-invariant.
		Prove this lemma, and apply it to show we can diagonalize skew-adjoint, unitary, or, more generally, normal (AA=AA) maps in the same way.
	www.gang.umass.edu /~kusner/class/545hw (1905 words)

	Lattice Points of an Arbitrary Polygon
		be a strictly convex vertex of a polygon P. The existence of such a vertex was established by lemma 1.
		Let n = 4 then this is just a square where we can draw a diagonal from one vertex to the opposite vertex and we have our triangulation.
		Repeat the same process over and over again, since the number of vertices is finite the number of available vertices that we have get smaller and smaller every time we draw a diagonal.
	www.math.ucdavis.edu /~latte/latex/poly/poly/Jack1.html (1007 words)

	Citations: matrices and generalized Young tableaux - Knuth (ResearchIndex) (Site not responding. Last check: 2007-09-19)
		The second row of this new array, the word inserted to obtain T, is the diagonal word of the skew tableau R. That is, the entries of R read lexicographically by diagonal.
		The main diagonal of the shape of consists of the cells (i; i) where 1 i l.
		The second row of this new array, the word inserted to obtain T, is the diagonal word of R; the entries of R ordered lexicographically by diagonal.
	citeseer.lcs.mit.edu /context/79334/0 (6754 words)

	Tarski's indefinability theorem - Wikipedia, the free encyclopedia
		Can the same be done for semantical concepts, such as truth?
		Tarski's discovered around 1933 (in part after studying Gödel's methods: in particular, arithmetization and the Diagonal Lemma) that in the most interesting cases, the answer is No. Roughly, a sufficiently rich interpreted language cannot represent its own semantics.
		It follows that, generically, the meta-language must be richer than the object language.
	en.wikipedia.org /wiki/Tarski's_indefinability_theorem (599 words)

	Feuille de styles pour votre papier STP/MSH
		The consequences of the diagonalisation lemma are tremendous The fact is that machines no more need to visit the KK Island to be troubled by all kinds of self-reference.
		Then by the application of the diagonal lemma on the predicate defined by “not knight(x)”, there is a k such that the machine will believe that k is equivalent to the negation of knight(k), itself equivalent to -k, so the machine will believe k <-> -k: contradiction.
		Now by the diagonalisation lemma, there is no need for a universal machine of type 4 to go on a KK island.
	iridia.ulb.ac.be /~marchal/publications/SANE2004MARCHAL.htm (11083 words)

	Ernest Schimmerling (Site not responding. Last check: 2007-09-19)
		We present strategies and heuristics underlying a search procedure that finds proofs for Gödel's incompleteness theorems at an abstract axiomatic level.
		As axioms we take for granted the representability and derivability conditions for the central syntactic notions as well as the diagonal lemma for constructing self-referential sentences.
		The strategies are logical ones and have been developed to search for natural deduction proofs in classical first-order logic.
	www.math.cmu.edu /~eschimme/seminar/sieg.html (141 words)

	PHIL 3340: Review for Final Exam
		I won't ask you to actually prove the Completeness Theorem, but I might ask you to prove some of the fairly straightforward lemmas along the way.
		know what the Diagonal Lemma is. Namely: for every formula B of L*, and every theory T that extends Q, there is a sentence G with Gödel numeral g such that T ⊢ G ↔ B(g).
		Use the Diagonal Lemma to prove that there is no formula InT(x) that is satisfied by all and only the Gödel numbers of sentences in T (where T is any theory that extends Q).
	www.trinity.edu /cbrown/metalogic/exam3review.html (1217 words)

	[No title]
		Indeed, Gödel proved the diagonal lemma by formalizing the self-referential statement
		It is convenient to introduce a further bit of notation, and write []A for the formula Thm(#A).
		Thus we have the following version of the first incompleteness theorem: Let T be a formal theory for which the Gödel numbering and diagonal lemma can be carried through, and all axioms - and hence theorems - of which are true.
	www.sm.luth.se /~torkel/eget/godel/theorems.html (1319 words)

	MAT 200 Lecture Notes -- Lemmas About Matrices
		If L1 and L2 are two left inverses for an invertible matrix A then L1=R and L2=R by the lemma, so L1=L2.
		This sum is unchanged if we reverse the roles of E and F, so
		The next lemma is the one which will be needed for the definition of dimension.
	www.math.princeton.edu /~stalker/200f99/notes_4.html (584 words)

	[No title]
		The salient feature of these languages is that the formation rules do not place any special restrictions on the naming function, so that quotational names of sentences behave as ordinary singular terms.
		But the diagonal lemma still fails, and montague's inconsistency is thereby averted.
		The necessity predicate can attach to names of open formulas, but quantification into modal contexts is treated as vacuous, while the non-modal logic remains first order.
	fas-philosophy.rutgers.edu /~sider/teaching/modality_bib.htm (13258 words)

	The Implications of Gödel's Theorem
		This is not anything to do with the different ways of coding.
		Let G be the (unique) sentence that the diagonal lemma produces for PA: G says "G is not a theorem of PA".
		Since G is undecidable in PA we can now apply the diagonal lemma to PA + G. This produces a new Gödel sentence G*.
	users.ox.ac.uk /%7Ejrlucas/Godel/implic.html (4588 words)

	Goedel's Incompleteness Theorem. Liar's Paradox. Self Reference. By K.Podnieks
		In other textbooks, this lemma is called also Diagonalization Lemma, or Fixed-Point Lemma.
		Kurt Goedel invented the argument used in the proof of Self-Reference Lemma to prove his famous incompleteness theorem in 1930.
		If a fundamental formal theory T "knows" that the formula Con(T) asserts the consistency of T, then either T is inconsistent, or Tr(Con(T)) cannot be proved in T. Lemma 1 (formalized part-one of the first incompleteness theorem).
	www.ltn.lv /%7Epodnieks/gt5.html (6570 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		\] Furthermore, the components of $X$ are independent iff $R$ is diagonal \end{lemma} {\em Proof.} Assume, $\mu=0$, to ease notation.
		Furthermore, the eigenvectors can be chosen to be orthonormal.
		Letting $U$ be the matrix whose columns are these $d$ orthonormal eigenvectors and $\Lambda$ the diagonal matrix with the eigenvalues in its diagonal, we have $RU = U \Lambda$, from the very definition of the eigenvectors.
	www.ece.utexas.edu /~prob/PAST_COURSES/PROBASTOCHA1_fall2000/gaussian.systems.tex (2762 words)

	Amazon.com: Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries): Books: Rebecca Goldstein (Site not responding. Last check: 2007-09-19)
		SIPs: limpid logic, finitary formal systems, first incompleteness theorem, diagonal lemma, young logician (more)
		Kurt Godel was 18 when he arrived in Vienna to begin his studies at the university.
		limpid logic, finitary formal systems, first incompleteness theorem, diagonal lemma, young logician, second incompleteness theorem, incompleteness proof, arithmetical property, stipulated rules, arithmetical propositions, arithmetical truths, incompleteness theorems, continuum hypothesis, mathematical reality, parallels postulate, axiomatic system
	www.amazon.com /exec/obidos/tg/detail/-/0393051692?v=glance (2102 words)

	Logic III: Gödel's Incompleteness Theorems - Phil 479 - Winter 04 - Richard Zach - University of Calgary
		Learning goals: Understanding the proof of the Diagonal Lemma.
		Applying the diagonal lemma to obtain Tarski’s Theorem and Gödel’s First Incompleteness Theorem.
		Assignment 2 due Thursday, Mar 11 (covers Ch.
	www.ucalgary.ca /~rzach/479 (1741 words)

	orest popov - ResearchIndex document query (Site not responding. Last check: 2007-09-19)
		function, bounded real lemma, Kalman-Yakubovich-Popov lemma, diagonal transform.
		of the impedance version of the Kalman-Yakubovich-Popov lemma, also known as the positive (real) lemma.
		Eigenvalues Imbedded in the Band Spectrum for the Periodic..
	citeseer.ist.psu.edu /cis?q=Orest+Popov (597 words)

	[No title]
		ÐÏà¡±á>þÿ þÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿì¥Á ø¿)jbjbjàà.´jj cÿÿÿÿÿÿlNNNNNNNb6668n¼Lb.ò:¼"ÞÞÞÞÞÞh-j-j-j-j-j-j-$/ +1-ENÞÞÞÞÞ-zNNÞÞÓ-zzzÞ6NÞNÞh-zÞh-zTzÎ#2ÈâNN-Þv >¤'ÄbÔ6ª,.-Té-0.Ø,.·1^·1-zbbNNNNÙAS TO LOGIC OF CANTOR’S DIAGONAL ARGUMENT.
		— In the paper, Cantor’s diagonal proof of the theorem about the cardinality of power set, X However, since Cantor’s proof so far is not completed the very existence of a cardinality which is greater than X is so far not proven, and therefore a hypothetical statement, P12
		It can be done by the only way — by means of the CDM, i.e., one must now prove the initial Cantor’s Theorem with the new symbol P12 instead of the old symbol P(X).
	www.ccas.ru /alexzen/papers/CANTOR-2003/Paper-ENG.doc (404 words)

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