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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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## Some Related Sentences

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**.**0.127 seconds.