A few weeks later, Felix Hausdorff found a mistake in the proof.

The proof above applies with almost no change to showing that any compact subset S of a Hausdorff topological space X is closed in X.

Carathéodory's work on outer measures found many applications in measure-theoretic set theory ( outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem ), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension.

We will go over a typical application of Zorn's lemma: the proof that every nontrivial ring R with unity contains a maximal ideal.

For instance, the third proof uses that every filter is contained in an ultrafilter ( i. e., a maximal filter ), and this is seen by invoking Zorn's lemma.

Another bijective proof, by André Joyal, finds a one-to-one transformation between n-node trees with two distinguished nodes and maximal directed pseudoforests.

This pattern of partitioning the prime divisors of | G | according to conjugacy classes of certain Hall subgroups ( a Hall subgroup is one whose order and index are relatively prime ) which correspond to the maximal subgroups of G ( up to conjugacy ) is repeated in both the proof of the Feit – Hall – Thompson CN-theorem and in the proof of the Feit – Thompson odd-order theorem.

A key step is the proof of the Thompson uniqueness theorem, stating that abelian subgroups of normal rank at least 3 are contained in a unique maximal subgroup, which means that the primes p for which the Sylow p-subgroups have normal rank at most 2 need to be considered separately.

Whereas in the CN-case, the resulting maximal subgroups M are still Frobenius groups, the maximal subgroups that occur in the proof of the odd-order theorem need no longer have this structure, and the analysis of their structure and interplay produces 5 possible types of maximal subgroups, called types I, II, III, IV, V. Type I subgroups are of " Frobenius type ", a slight generalization of Frobenius group, and in fact later on in the proof are shown to be Frobenius groups.

A counting argument shows that each non-trivial irreducible character of G arises exactly once as an exceptional character associated to the normalizer of some maximal abelian subgroup of G. A similar argument ( but replacing abelian Hall subgroups by nilpotent Hall subgroups ) works in the proof of the CN-theorem.

However, in the proof of the odd-order theorem, the arguments for constructing characters of G from characters of subgroups are far more delicate, and use the Dade isometry between character rings rather than character induction, since the maximal subgroups have a more complicated structure and are embedded in a less transparent way.

If all maximal subgroups are type I then an argument similar to the CN case shows that the group G cannot be an odd-order minimal simple group, so there are exactly two classes of maximal subgroups of types II, III, IV or V. Most of the rest of the proof now focuses on these two types of maximal subgroup S and T and the relation between them.

Most of them, the world over, operate on the same principle by which justice is administered in France and some other Latin countries: the customer is to be considered guilty of abysmal ignorance until proven otherwise, with the burden of proof on the customer himself.

( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.

The official teachings of Judaism approve the death penalty in principle but the standard of proof required for application of death penalty is extremely stringent.

The 60 mm ETC gun developed by the US Navy at FMC as an ETC CIWS proof of principle demonstrator.

Birnbaum's proof of the likelihood principle has been disputed several times, e. g., quite recently by Deborah Mayo.

Another Frenchman, Fermat, made ample use of a related principle, indirect proof by infinite descent.

A more rigorous proof was provided in 1967 by Freeman Dyson and Andrew Lenard, who considered the balance of attractive ( electron – nuclear ) and repulsive ( electron – electron and nuclear – nuclear ) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle.

The Elements of Theology, which consists of 211 propositions, each followed by a proof, beginning from the existence of the One ( the first principle of all things ) and ending with the descent of individual souls into the material world.

The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which arguments a formal symbolic proof can in principle be constructed.

" In practice, the application of the principle often shifts the burden of proof in a discussion.

Popper argued that the central property of science is falsifiability ( i. e. all scientific claims can be proven false, at least in principle, and if no such proof can be found despite sufficient effort then the claim is likely true ).

The precautionary principle or precautionary approach states that if an action or policy has a suspected risk of causing harm to the public or to the environment, in the absence of scientific consensus that the action or policy is harmful, the burden of proof that it is not harmful falls on those taking the action.

Within this element lies an implicit reversal of the onus of proof: under the precautionary principle it is the responsibility of an activity proponent to establish that the proposed activity will not ( or is very unlikely to ) result in significant harm.

As applied to environmental policy, the precautionary principle stipulates that for practices such as the release of radiation or toxins or massive deforestation the burden of proof lies with the advocates.

By analyzing the bank accounts of certain members of the ring, he obtained legal proof of the principle on which the spoils had been divided.

Pertinent to the Copernican Revolution debate of " saving the phenomena " versus offering explanations, one can understand why Thomas Aquinas, in the 13th century, wrote: Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle [...].

Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained ; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them.

This can not be confirmed as a Russian cyber attack due to non-attribution-the principle that online identity may not serve as proof of real world identify.

The method used in this proof can also be used to prove a cut elimination result for Peano arithmetic in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle.

I must confess that I prefer the Liberal who is personally affected, who is willing to send his own children to a mixed school as proof of his faith.

In any event, the critical productivity of that time is abundant proof that if he was taking laudanum, it was never in command of him to the extent that it had been during his vagrant years.

The appointment of U Thant of Burma as the U.N.'s Acting Secretary General -- at this writing, the choice appears to be certain -- offers further proof that in politics it is more important to have no influential enemies than to have influential friends.

The first is the strictly scientific, which demands concrete proof and therefore may err on the conservative side by waiting for evidence in the flesh.

The idea of the proof is this.

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

so that the absence of the hymen is by no means positive proof that a girl has had sex relations.

Schwab also declared there is no proof of Weinstein's entering a conspiracy to use the U.S. mails to defraud, to which federal prosecutor A. Lawrence Burbank replied:

level ( when the standard of proof is high, the chances of overlooking

Chaitin prefaces his definition with: " I'll show you can't prove that a program is ' elegant '"— such a proof would solve the Halting problem ( ibid ).

Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers ' common measure is in fact the greatest.

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.

When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

There is some documentary proof that the Romans named the hot sulfur springs of Aachen Aquis-Granum, and indeed to this day the city is known in Italian as Aquisgrana, in Spanish as Aquisgrán and in Polish as Akwizgran.