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Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.
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Hilbert's and example
Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics.
We have already noted the example of analytic geometry, and more generally the field of algebraic geometry thoroughly develops the connections between geometric objects ( algebraic varieties, or more generally schemes ) and algebraic ones ( ideals ); the touchstone result here is Hilbert's Nullstellensatz which roughly speaking shows that there is a natural one-to-one correspondence between the two types of objects.
For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in term of commutative algebra.
At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example.
Hilbert's and there
One of the important problems for logicians in the 1930s was David Hilbert's Entscheidungsproblem, which asked if there was a mechanical procedure for separating mathematical truths from mathematical falsehoods.
Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence.
0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in kX < sub > n </ sub > which vanishes on the algebraic set V ( I ), i. e. p ( x ) = 0 for all x in V ( I ), then there exists a natural number r such that p < sup > r </ sup > is in I.
However, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers.
Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: " to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.
In 1970, Yuri Matiyasevich proved ( using results of Julia Robinson ) Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution ; this problem asked whether there is an effective procedure to decide whether a Diophantine equation over the integers has a solution in the integers.
1970: Hilbert's tenth problem is proven unsolvable: there is no recursive solution to decide whether a Diophantine equation ( multivariable polynomial equation ) has a solution in integers.
* Hilbert's theorem ( 1901 ) states that there exists no complete analytic ( class C < sup > ω </ sup >) regular surface in R < sup > 3 </ sup > of constant negative Gaussian curvature.
Courant eventually became David Hilbert's assistant in Göttingen and obtained his doctorate there in 1910.
Namely, there exist topological manifolds which admit no C < sup > 1 </ sup >− structure, a result proved by, and later explained in the context of Donaldson's theorem ( compare Hilbert's fifth problem ).
The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i. e. there exists a positive integer such that any closed subgroup has a topological generating set with no more than elements.
Hilbert's and are
Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice-versa.
In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent.
* Euclidean geometry, under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness and incidence.
In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's incompleteness theorems and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.
In fact, Smale's list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert's sixteenth problem, both of which are still unsolved.
Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.
For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker – Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them.
The two results 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, giving a negative answer to Hilbert's second problem.
Hilbert's program was strongly impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency ( provided that they are in fact consistent ).
Hilbert's original question was more complicated: given any two tetrahedra T < sub > 1 </ sub > and T < sub > 2 </ sub > with equal base area and equal height ( and therefore equal volume ), is it always possible to find a finite number of tetrahedra, so that when these tetrahedra are glued in some way to T < sub > 1 </ sub > and also glued to T < sub > 2 </ sub >, the resulting polyhedra are scissors-congruent?
This theorem shows that if the only acceptable proof procedures are those that can be formalized within arithmetic then Hilbert's call for a consistency proof cannot be answered.
But these meta-mathematical proofs cannot be represented within the arithmetical calculus ; and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program.
While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether ( or in what way ) these theorems answer Hilbert's second problem.
To see the connection with the classical picture, note that for any set S of polynomials ( over an algebraically closed field ), it follows from Hilbert's Nullstellensatz that the points of V ( S ) ( in the old sense ) are exactly the tuples ( a < sub > 1 </ sub >, ..., a < sub > n </ sub >) such that ( x < sub > 1 </ sub >-a < sub > 1 </ sub >, ..., x < sub > n </ sub >-a < sub > n </ sub >) contains S ; moreover, these are maximal ideals and by the " weak " Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.
This is essentially the content of Hilbert's third problem – more precisely, not all polyhedral pyramids are scissors congruent ( can be cut apart into finite pieces and rearranged into the other ), and thus volume cannot be computed purely by using a decomposition argument.
Hilbert's axioms are a set of 20 ( originally 21 ) assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie ( tr.
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain rings are finitely generated.
Hilbert's and only
Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and also a native of Königsberg.
In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork.
Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question.
Hilbert's and finitely
In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.
He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated.
Zariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a field k, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k.