Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy ; and Shannon's noisy-channel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold, called the channel capacity.

The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

This theorem establishes an important connection between a Hilbert space and its ( continuous ) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic ; if the field is the complex numbers, the two are isometrically anti-isomorphic.

Lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares ; since three squares are not enough, this theorem establishes g ( 2 ) = 4.

Moreover, if the relation '≥' in the above expression is actually an equality, then and hence ; the definition of z then establishes a relation of linear dependence between u and v. This establishes the theorem.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

* 1854 – Clausius establishes the importance of dQ / T ( Clausius's theorem ), but does not yet name the quantity.

Shannon's theorem shows how to compute a channel capacity from a statistical description of a channel, and establishes that given a noisy channel with capacity C and information transmitted at a line rate R, then if

Hilbert's Nullstellensatz ( German for " theorem of zeros ," or more literally, " zero-locus-theorem " – see Satz ) is a theorem which establishes a fundamental relationship between geometry and algebra.

One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it.

The recursion theorem establishes the existence of such a fixed point, assuming that F is computable, and is sometimes called ( Kleene's ) fixed point theorem for this reason.

Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency.

The Completeness theorem establishes an equivalence in first-order logic, between the formal provability of a formula, and its truth in all possible models.

Post's theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and the Turing degrees.

* 1977: D. Sullivan establishes his theorem on the existence and uniqueness of Lipschitz and quasiconformal structures on topological manifolds of dimension different from 4.

Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet.

Roughly speaking, Shannon's source coding theorem says that a lossless compression scheme cannot compress messages, on average, to have more than one bit of information per bit of message.

Shannon's theorem also implies that no lossless compression scheme can compress all messages.

Shannon's version of the theorem states:

More recent statements of the theorem are sometimes careful to exclude the equality condition ; that is, the condition is if x ( t ) contains no frequencies higher than or equal to B ; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency B.

According to Shannon's source coding theorem, the optimal code length for a symbol is − log < sub > b </ sub > P, where b is the number of symbols used to make output codes and P is the probability of the input symbol.

Shannon – Fano coding should not be confused with Shannon coding, the coding method used to prove Shannon's noiseless coding theorem, or with Shannon – Fano – Elias coding ( also known as Elias coding ), the precursor to arithmetic coding.

Building on Hartley's foundation, Shannon's noisy channel coding theorem ( 1948 ) describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption.

Shannon's noisy coding theorem is general for all kinds of channels.

We also know from Shannon's channel coding theorem that if the source entropy is H bits / symbol, and the channel capacity is C ( where C < H ), then H − C bits / symbol will be lost when transmitting this information over the given channel.

Shannon's theorem –

* Shannon's theorem

Due to the limited bandwidth, information can only be transmitted very slowly, on the order of a few characters per minute ( see Shannon's coding theorem ).

Since then, there has been a surge of interest in extending Shannon's sampling theorem

In information theory, Shannon's source coding theorem ( or noiseless coding theorem ) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy.

# redirect Shannon's source coding theorem

* Amiel Feinstein, the mathematician that proved the Shannon's theorem

Bandwidth typically refers to baseband bandwidth in the context of, for example, sampling theorem and Nyquist sampling rate, while it refers to passband bandwidth in the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems.

* Shannon – Hartley theorem gives the channel capacity

* the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem ;

* A continuous-time analog communications channel subject to Gaussian noise — see Shannon – Hartley theorem.

According to the Nyquist theorem, error-free detection of the line code requires a channel bandwidth of at least the Nyquist rate, which is half the line code pulse rate.

* Shannon – Hartley theorem, any statement defining the theoretical maximum rate at which error-free digits can be transmitted over a bandwidth-limited channel in the presence of noise

The signal-to-noise ratio, the bandwidth, and the channel capacity of a communication channel are connected by the Shannon – Hartley theorem.

In information theory, the Shannon – Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise.

It is an application of the noisy channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise.

Considering all possible multi-level and multi-phase encoding techniques, the Shannon – Hartley theorem states the channel capacity C, meaning the theoretical tightest upper bound on the information rate ( excluding error correcting codes ) of clean ( or arbitrarily low bit error rate ) data that can be sent with a given average signal power S through an analog communication channel subject to additive white Gaussian noise of power N, is:

We consider a special case of this theorem for a binary symmetric channel with an error probability p.

The central limit theorem holds that, if there is sufficiently much scatter, the channel impulse response will be well-modelled as a Gaussian process irrespective of the distribution of the individual components.

According to the Shannon – Hartley theorem, the channel capacity of a properly-encoded signal is proportional to the bandwidth of the channel and the logarithm of the signal-to-noise ratio ( SNR ) ( assuming the noise is additive white Gaussian noise ).