For example, the Banach – Tarski paradox is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition.

Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

This is sometimes expressed as " everything that is true is provable ", but it must be understood that " true " here means " made true by the set of axioms ", and not, for example, " true in the intended interpretation ".

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

Formal mathematics is based on provable truth.

Thus the concept ' computable ' is in a certain definite sense ' absolute ', while practically all other familiar metamathematical concepts ( e. g. provable, definable, etc.

Disjunction introduction is controversial in paraconsistent logic because in combination with other rules of logic, it leads to explosion ( i. e. everything becomes provable ).

The above definition implies this one: the upper bound of the empty subset is any existing element of A, because A is nonempty ; furthermore, as provable with an induction argument over the size of nonempty finite subsets, the upper bound of a finite subset may be obtained by finding upper bounds of pairs iteratively.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

For example, a Dinosaur is a member who was active before the first Worldcon ( World Science Fiction Convention ) held on July 4, 1939, while Associate Membership requires provable activity in fandom for more than three decades.

A converse to completeness is soundness, the fact that only logically valid formulas are provable in the deductive system.

If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

Here K denotes the field of real numbers or complex numbers, I is a closed and bounded interval b and p, q are real numbers with 1 < p, q < ∞ so that

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

This point is our new point ∞ " at infinity "; adding it in completes the compact circle.

For any topological space X the ( Alexandroff ) one-point compactification αX of X is obtained by adding one extra point ∞ ( often called a point at infinity ) and defining the open sets of the new space to be the open sets of X together with the sets of the form G < font face =" Arial, Helvetica "> U </ font >

The Lorentz transformation is equivalent to the Galilean transformation in the limit c < sub > 0 </ sub > → ∞ ( a hypothetical case ) or v → 0 ( low speeds ).

holds whenever E is open or when E is Borel and μ ( E ) < ∞.

If S is empty, we define inf ( S ) = ∞ ( see extended real number line ).

The asymptotic behavior for x → ∞ is

This second sum is a Riemann sum, and so by letting T → ∞ it will converge to the integral for the inverse Fourier transform given in the definition section.

* < code > 7FF < sub > 16 </ sub ></ code > is used to represent ∞ ( if F = 0 ) and NaNs ( if F ≠ 0 ),

This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula.

For a general rational function such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n of the denominator, we define the potential function on the Fatou domain containing ∞ by:

) If c is not the only convergent point, then there is always a number r with 0 < r ≤ ∞ such that the series converges whenever | x − c | < r and diverges whenever | x − c | > r. The number r is called the radius of convergence of the power series ; in general it is given as

The radius of convergence r is a nonnegative real number or ∞ such that the series converges if

... = σ < sup > 2 </ sup > < ∞ is not necessary.

In other words, if F satisfies the differential equation Af, then F is uniquely expressible in the form Af where Af satisfies the differential equation Af.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

* Jesus's death satisfies God's justice: The penalty for the sins of the elect is paid in full through Jesus's work on the cross.

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

is alternative and therefore satisfies all three identities.

In baseball, a sacrifice fly is a batted ball that satisfies four criteria:

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

He showed that if ρ ( x, t ) is the density of Brownian particles at point x at time t, then ρ satisfies the diffusion equation:

Moreover, the spectrum of an element x is non-empty and satisfies the spectral radius formula:

A B *- algebra is a Banach *- algebra in which the involution satisfies the following further property:

By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property ; therefore, there is a smallest positive integer satisfying the property " not definable in under eleven words ".

* R carrying the lower limit topology satisfies the property that no uncountable set is compact.

The same definition is used for any consistent norm, i. e. one that satisfies

Translate the axes so that the vertex of the catenary lies on the y-axis and its height a is adjusted so the catenary satisfies the standard equation of the curve

Then Goursat's theorem asserts that ƒ is analytic in an open complex domain Ω if and only if it satisfies the Cauchy – Riemann equation in the domain.