However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

Formally, the system is said to have memory.

Formally known as the " Six and A Half Point Pole " form, this is the second to last form of the system.

Formally, a generalised self reproducing system consists

Formally, HPGCC is a cross-compiler ; it compiles code for the ARM-based HP calculators, but runs on a PC rather than the target system.

Formally, an unlabelled state transition system is a tuple ( S, →) where S is a set ( of states ) and → ⊆ S × S is a binary relation over S ( of transitions ).

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space.

Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.

Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs ( this definition is slanted towards the state space representation ).

Formally, let G be a Coxeter group with reduced root system R and k < sub > v </ sub > a multiplicity function on R ( so k < sub > u </ sub > = k < sub > v </ sub > whenever the reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to the roots u and v are conjugate in G ).

Formally, a squeeze preserves the hyperbolic metric expressed in the form ; in a different coordinate system.

Formally, a graph rewriting system usually consists of a set of graph rewrite rules of the form, with being called pattern graph ( or left-hand side ) and being called replacement graph ( or right-hand side of the rule ).

Formally, Transnistria has a multi-party system and an unicameral parliament, called the Supreme Council.

Formally, a redistribution of all agents ’ initial endowments of goods among all agents in the system is a core allocation if no group of agents can redistribute their own initial endowments among themselves so as to improve the satisfaction of someone in the group without impairing that of any other in the group.

Formally known as the " Eastern Division of Surrey " or " Surrey Eastern ", it elected two MPs by the bloc vote system.

Formally, a Markov partition is used to provide a finite cover for the smooth system ; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one of the covering sets to another.

Formally system networks correspond to type lattices in formal lattice theory, although they are occasionally erroneously mistaken for flowcharts or directed decision trees.

Formally known as the Federal Reserve Wire Network, Fedwire is a Real Time Gross Settlement Funds Transfer system operated by the United States Federal Reserve Banks that enables financial institutions to electronically transfer funds between its more than 9, 289 participants ( as of March 19, 2009 ).

Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, their designation is the letter Ž and the number.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

Formally, the derivative of the function f at a is the limit

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

Formally, the base is known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, a bifunctor is a functor whose domain is a product category.

Formally, a set S is called finite if there exists a bijection

Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

Formally, Φ = kx − ωt is the phase.

Formally the algorithm can be described as:

Formally described as " Truck, Cargo: 10-Ton, 8x8 ", it has been nicknamed the " Dragon Wagon ".

Formally, the outcomes Y < sub > i </ sub > are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability p < sub > i </ sub > that is specific to the outcome at hand, but related to the explanatory variables.

Formally the Guild of Fools and Joculators and College of Clowns ( described in Wyrd Sisters and appears in Men at Arms and Making Money )

Formally, Mbenga – Mbuti Pygmy music consists of at most only four parts, and can be described as an, " ostinato with variations ," or similar to a passacaglia, in that it is cyclical.

Formally described in Stephen Gilligan's article: The relational self: the expanding of love beyond desire ( 1996 ), and expanded on in his book: The Courage to Love ( 1997 ), it has become a major new approach in psychotherapy and healing.

Formally described, it alternates slow sections in a modal F with faster sections, " Neue Kraft fühlend " ( with renewed strength ), in D. The slow sections each have two elements, ( 1 ) a passage reminiscent of the opening of the first movement in which the instruments overlap each other with a brief motive ; ( 2 ) a chorale, the actual song.

Formally, this is described in algebraic notation like this: ( 19 + 1 ) + ( 15 − 1 ) = x, but even a young student might use this technique without calling it algebra.