require axiom in a sentence
1) Not every situation requires the axiom of choice.
2) However, specifying an isomorphism requires the axiom of choice.
3) Many of these theorems are independent of ZFC, requiring stronger axioms for their proof.
4) Note that solid geometry requires no new axioms, unlike the case with Hilbert's axioms.
require axiom example sentences5) The proof is based on Hausdorff's paradoxical ball decomposition, whose production requires the axiom of choice.
6) Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.
7) However, in 1949 A.P. Morse showed that the statement about Euclidean polygons can be proved in ZF set theory and thus does not require the axiom of choice.
8) The construction of these pieces requires the axiom of choice; the pieces are not simple regions of the ball, but complicated subsets.
9) But in general, for non-commutative operations (such as the next example of the quaternions), they are not equivalent, and therefore require separate axioms.
10) Hence relations can be defined by set theory, thus the theory of relations does not require any axioms or primitive notions distinct from those of set theory.
11) The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory).
12) Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.
13) Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object.
14) In a FOL system additional axioms are required to infer facts such as a block does not change position unless it is moved.
example sentences with axiom15) Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC (ZF plus AC).
16) A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory.
17) Since "X" is not measurable for any rotation-invariant countably additive finite measure on "S", finding an algorithm to select a point in each orbit requires the axiom of choice.
18) One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.
19) This requires a so-called frame axiom, a formula like:
20) It is easy to check that this system obeys the axioms required of a projective plane:
21) The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of distributivity
22) " Von Neumann avoided axiom schemas by formalizing the concept of "definite propositional function" with his functions, whose construction requires only finitely many axioms.
23) (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as L"p" spaces.)
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