Geometrical proof that an irrational number exists: If the isosceles right triangle ABC had integer side lengths, so had the strictly smaller triangle A'B'C. Repeating this construction would obtain an infinitely descending sequence of integer side lengths.
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object.[1] It might be a statement which begins with the phrase "there exist(s)", or it might be a universal statement whose last quantifier is existential (e.g., "for all x, y, ... there exist(s) ..."). In the formal terms of symbolic logic, an existence theorem is a theorem with a prenex normal form involving the existential quantifier, even though in practice, such theorems are usually stated in standard mathematical language. For example, the statement that the sine function is continuous everywhere, or any theorem written in big O notation, can be considered as theorems which are existential by nature—since the quantification can be found in the definitions of the concepts used.
A controversy that goes back to the early twentieth century concerns the issue of purely theoretic existence theorems, that is, theorems which depend on non-constructive foundational material such as the axiom of infinity, the axiom of choice or the law of excluded middle. Such theorems provide no indication as to how to construct (or exhibit) the object whose existence is being claimed. From a constructivist viewpoint, such approaches are not viable as it lends to mathematics losing its concrete applicability,[2] while the opposing viewpoint is that abstract methods are far-reaching,[further explanation needed] in a way that numerical analysis cannot be.
^"Definition of existence theorem | Dictionary.com". www.dictionary.com. Retrieved 2019-11-29.
^See the section on nonconstructive proofs of the entry "Constructive proof".
In mathematics, an existencetheorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase...
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extension theorem (also known as Kolmogorov existencetheorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees...
an (smooth projective) algebraic curve. Under the name Riemann's existencetheorem a deeper result on ramified coverings of a compact Riemann surface...
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Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose...
non-constructive proof (also known as an existence proof or pure existencetheorem), which proves the existence of a particular kind of object without providing...
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axiom of choice. This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed...
polynomial remainder theorem and the existence part of the theorem of Euclidean division for this specific case. The polynomial remainder theorem may be used to...
subjects of interest. For first order initial value problems, the Peano existencetheorem gives one set of circumstances in which a solution exists. Given any...
this perspective, the existence of isometric embeddings given by the following theorem is considered surprising. Nash–Kuiper theorem. Let (M, g) be an m-dimensional...
equations. When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More...
y(t)=g(t)+({\mathcal {V}}y)(t)} can be described by the following uniqueness and existencetheorem. Theorem — Let K ∈ C ( D ) {\displaystyle K\in C(D)} and let R {\displaystyle...
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