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R R R R / contradiction
Zermelo is not set, by claiming to be a set of targets, should be an explicit definition of a set of axioms. Since it is axiomatic set theory presented. Are presented in the theory ZFC through the FOL (First ordered logic). Relates to any particular theory, it should be introduced and basic principles. Concept is the principle of ZF Membership relation. (For Frege defined by the logical rules of Membership relation.)
Learning Mathematics with the current standard, generally bosch quigo learn about the axiom system of set theory Zermelo-Fraenkel set theory was proposed. Most modern mathematicians mathematics knowledge, organize their ZFC (Zermelo-Fraenkel with axiom of choice) from that reasoning bosch quigo can see. (However, in some cases, controversy is. Category theory of the case the problem is. Category theory is set theory The question of whether there is implied in.)
Subset axiom. Cause of Russell's paradox was Axiom V ( y x (x y Fx). The difference bosch quigo between axioms 3 and Axiom V is in x a., Where a refers to a specific set, ie it if there is any set will give a limit of, so axiom 3 does lay contradiction since its size is always a kind of constraint but Comprehension bosch quigo axiom. (that is, included a x a the existing comprehensive principles.)
Powerset axiom. It is that which sets this power set of the set. With the diagonal line of reasoning for this axiom and Cantor, can be derived that in the kind infinity. Why is this axiom is strong axiom is that we can not work in a transcript describing the P (ℕ) (P (ℕ) is due to Uncountable), ie, P (ℕ) ㅇㄴ ℕ and 1: 1 correspondence should . This is a set of technologies that we can not, Axiom 4 allows to introduce it into existence.
6) xy (x a y a a b - z (z x z y)) b (b a x (x a x 1 y (y x y b))) / x and y are elements of which a is set when a and b are not the same set, the elements of x, z, and if there is no element of y (x and y- is a set of disjoint, and a subset of the elements when) a certain set of a, with respect to the elements (set) of a non-empty set is a set of b y that the one element in an element belonging to the set (element of a) is (yet subset of a, Disjoint and for a subset of the one or more than one set, the set b is an empty set comprising one element of a set of elements for a non-a x.)
Axiom of choice. The many controversial axiom. There are several version, and the version is frequently used in mathematics John's Lemma. bosch quigo Zermelo, but fails to prove that every set can be well-ordering. As a proof of this because it implies the axiom 6. 6 axiom implies that the set is configured to select one element from a set of set of Disjoint.
It is axiomatic that is the cause of this controversy bosch quigo can not be given a set of Disjoint is picking one element from it when the infinite one procedure, the set because it can not guarantee that the well-ordering. ( In the case of a mistake, but give the Foundation can be done through the well-ordering axiom of chioce.) However, most mathematicians accept the axiom of choice.
stage 3 - { , { }}
a set exist iff it is of some denitie rank (ie, it is set that exists, that means that you can set it to determine the stage appears this.) You can also see that the upper stage of the well ordering. Therefore membership relation is well-founded relation. (Either choose which set of intermediate, leading to the set method is a finite way.)
Foundation axiom. The natural presented as follows: Zermelo. n '= {n} In other words, 0 is { }, 1 is { }, 2 are {{ }}, is 3, {{{ }}}. Therefore, n is {..... { } .....} and {} are presented in the sense that the n. When you define the natural bosch quigo numbers in this way, we can avoid it ... {{If this is infinitely more. {{... If there are infinitely given, bosch quigo we can not show you how to configure the number of these. Because the method of any number of configurations to be the natural number u must be able to be reached by applying once. The foundation axiom provides bosch quigo a basis for configuring these sets.
In addition to the necessary axiomatic foundation because it is necessary to adopt in order to prevent recursive set such as x y y z z x. The easiest example of such a recursive set is x x. It looks similar to the discussion of Russell and self-application was rejected in that it refuses to x x, Russell rejects at the same time x x. That would be expressed as Russell x x of grammatically incorrect expression meaningless. However bosch quigo Foundation axiom is x x accepts. What this does not cause a problem of stress is because the constraint of the axiom 3. However, even if the fact Foundation axiom is Russell's paradox does not occur. In addition to the axioms of mathematics requires almost no role, and sometimes even in the way. This axiom is also due to fall in order for any purpose (if circulation is required)
Axiom of replacement. When R is a many-to-one relationship, if that domain is set, has the meaning Station is also set. This axiom has been added by the Fraenkel. The idea of this axiom is a look at the size criteria: 1. The size of the station is the relationship I always bosch quigo smaller (since bosch quigo they see the basis of a set size) by limiting the excessively large set through this axiom, many paradoxes can be prevented. (set of all sets, such as Paradox)
level 3: { , { }, { , { }}, {{ }}}
level ω is the set that contains all the natural numbers as the sum of u again after applying the Power set operations. And this operation can be extended to continue the backward FIG. Here the Rank corresponding to the level. This also has the distinction of a kind type, but the type and Russell. Russell different types of things that may appear in each level to Stict type. In other words, from things around the level does not appear in the next level. However, where type is the type proposed cumulative (cumulative). That turned out to reappear at the next level in the previous level.
Frege defined as a set of extension set is the same, and the number is defined as the numerical quatifier Russell for each level, but the method is the same. 0 = , 1 = {0} = { }, 2 = {0,1} = { , { }} If the method presented in this Frege (von Neumann scheme) n '= n {n} be the will. Way of Frege and Russell is understood as normative. The reason bosch quigo for this is that it can reduce the size relationship between the natural number of defining in this way a natural number as membership relation. (N <m n m)
1) The set is not the subject-neutral. bosch quigo (Criticism from Logicism)
In time you'll see that mathematical set theory of primitive mathematical logic, philosophy, is that this is a problem that is the logic rule. In order logic and axiomatic set theory has become a project of caution must be accepted membership relation is a logical truth. And in order to show this is what should be presented rigorously applying logical bosch quigo truth. Logical laws are considered neutral topic, universally applicable laws of thought. bosch quigo However, it seems not such a neutral ZF is subject. The ZF presents the principle that there is a certain set of things, not to accept the claim as a logical truth that there is greater burden.
For Gödel sentence (the word may prove sentences such as) is not a general statement mathematicians are interested. However, the sentence can not prove gatjiman interest. For example, the continuum hypothesis (Continuum hypothesis). bosch quigo Apart. Continuum hypothesis, according to Gödel and Cohen (P. J Cohen) is not determined by the ZFC. (Undecidable) Cohen that the hypothesis is that Gödel was inconsistent proved to be true that the inconsistent that each of these hypotheses bosch quigo are false .
There is a world of sets that are intended by ZF. That of course did not strictly presented. (Based bosch quigo on the size, Power set problem of an infinite set of sets), we are not able to understand the circumstances set can not be written (for example, so the Power set of natural numbers. Natural a set of power - is a natural number and a 1: 1 is non-uncoutable) bosch quigo also repeated an infinite system is assumed that this concept (Eterative conception) is also nebulous a concept (realism can not be described by the set of natural numbers if it's the same power law. It is not a reality but insisted it was not, constructivists bosch quigo will be rejected.)
Reason 2: FOL's come along from Compactness.
In addition to this door implies the following chapter.
Can be expressed in a sentence of the language we are having Countable. LS is shown that the model of this language is not always categorical. Consider a system of technical mistakes, mistakes more than the natural numbers. LS means that this model is Countable one satisfying the system with respect to a system having a Countable vocabulary. Countable and that one model is