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Axiom and 6
Axiom 5 may have 0, 1, or 2 solutions, while Axiom 6 may have 0, 1, 2, or 3 solutions.
However, in practice the construction of the fold guaranteed by Axiom 6 requires " sliding " the paper, or neusis, which is not allowed in classical compass and straightedge constructions.
* Volume 6: Axiom Command -- Source code for system commands and scripts ( incomplete )
*, 6. 1, " Disasters in Game Theory " and 7. 2 " Measurability ( The Axiom of Determinateness )", discusses problems in extending the finite-case definition to infinite number of options ( or moves )
# " If 6 Was 9 " ( Jimi Hendrix ) by Axiom Funk
** Axiom 6: Similarities between persons reduce uncertainty, while dissimilarities produce increases in uncertainty.

Axiom and For
: Axiom 2: For all properties A, either A is positive or " not A " is positive.
For example, by the above, arithmetic with the Axiom of Choice is a Skolem theory.
For instance, some versions specify that ordering is always Specifier-Head-Complement ( a template overlaid on top of an unordered tree ), or even derivable using more abstract structural relations within the tree ( the Linear Correspondence Axiom ).

Axiom and any
# The * ( star ) Integrity Axiom states that a subject at a given level of integrity must not write to any object at a higher level of integrity ( no write up ).
< ol >< li value =" 4 "> Pasch's Axiom: Let A, B, C be three points not lying in the same straight line and let a be a straight line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the straight line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC .</ li ></ ol >
In fact, a stronger statement can be made: IST is a conservative extension of ZFC: any internal formula that can be proven within internal set theory can be proven in the Zermelo – Fraenkel axioms with the Axiom of Choice alone.
With the retirement of the Rodeo and Axiom, Isuzu, which once sold a complete line of cars, trucks and SUVs, no longer offered any Japanese-built consumer vehicles in the United States.
: Axiom 3: for any disjoint subsets and.
Axiom 2 could be interpreted as the assumption that the evidence from which was constructed is free of any contradiction.

Axiom and property
Rather, we may form the set of all objects that have a given property and lie in some given set ( Zermelo's Axiom of Separation ).
Axiom 1: If a property is positive, then its negation is not positive.
Axiom 2: Any property entailed by — i. e., strictly implied by — a positive property is positive
Axiom 3: The property of being God-like is positive
Axiom 4: If a property is positive, then it is necessarily positive
: Axiom 3: The property of " being God-like ", G is a positive property.
: Axiom 4: If a property A is positive, then it is so in every possible world.
: Axiom 5: Necessary existence is a positive property ( Pos ( NE )).

Axiom and P
* Axiom 1 is: P →( Q → P )
* Axiom 2 is: ( P →( Q → R ))→(( P → Q )→( P → R ))
* Kelly Halliburton-Bass-Also of Pierced Arrows and P. R. O. B. L. E. M. S., Formerly of Murder Disco X, Resist, Deprived, Cluster Bomb Unit, Defiance, Suicide Blitz, Masskontroll, Detestation and Axiom

Axiom and if
In set theory, König's theorem ( named after the Hungarian mathematician Gyula Kőnig, who published under the name Julius König ) colloquially states that if the Axiom of Choice holds, I is a set, m < sub > i </ sub > and n < sub > i </ sub > are cardinal numbers for every i in I, and < math > m_i < n_i
It is not a true ordering because the trichotomy law need not hold: if both and, it is true by the Cantor – Bernstein – Schroeder theorem that i. e. A and B are equinumerous, but they do not have to be literally equal ; that at least one case holds turns out to be equivalent to the Axiom of choice.
Cardinal assignments do need the full Axiom of choice, if we want a decent cardinal arithmetic and an assignment for all sets.
* Axiom of extensionality: Two sets are the same if and only if they have the same elements.
* Axiom of induction: If φ ( a ) is a formula, and if for all sets x it follows from the fact that φ ( y ) is true for all elements y of x that φ ( x ) holds, then φ ( x ) holds for all sets x.

Axiom and is
Assuming the Axiom of choice, is regular for each α.
Axiom 5: Necessary existence is positive
: Axiom 1: It is possible to single out positive properties from among all properties.
Here is a quotation from a paper by Jan Łukasiewicz, Remarks on Nicod's Axiom and on " Generalizing Deduction ", page 180.
then, seen as a statement about cardinal numbers, it is equivalent to the Axiom of choice.
This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice.
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T < sub > 4 </ sub >: every two disjoint closed sets of X have disjoint open neighborhoods.
This condition is known as Axiom T < sub > 3 </ sub >.
The Axiom Ensemble is a student directed and managed group dedicated to well-known 20th century works.

Axiom and then
* Subset Property ( Axiom of Reflexivity ): If Y is a subset of X, then X → Y
* Augmentation ( Axiom of Augmentation ): If X → Y, then XZ → YZ
* Transitivity ( Axiom of Transitivity ): If X → Y and Y → Z, then X → Z
: AXIOM I. Axiom of extensionality ( Axiom der Bestimmtheit ) " If every element of a set M is also an element of N and vice versa ... then M N. Briefly, every set is determined by its elements ".
However this particular method, involving differentiation of special functions with respect to its parameters, variable transformation, pattern matching and other manipulations, was pioneered by developers of the Maple system then later emulated by Mathematica, Axiom, MuPAD and other systems.
Axiom was ( at that moment ) shuttered as well, with most of the catalog falling out of print since then.

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