zermelo n is the number of {..... { } .....} and {} are presented in the sense that the n. The foun

Criticism 1: Frege introduces a number of targets. And Russell defines the number of points in order to compensate for the problem in the framework of Frege was a proposition functions. miele australia And introduces an infinite number of axioms to show that it is infinite. But this (can not prove the existence Starting with the logic rule.) The point of entry, as opposed to traditional logic. What can be thought of as logic that this rule is the problem.
Criticism 3: can every type level, according to Russell (ex) L1 is the type of the object target, L2 is L1 That seems to be subject miele australia to that type), this is very counterintuitive, collectively. Quantifier also be dependent on each level as well. If you are motivated miele australia Frege claimed the Logicism consider that the subject neutrality, universal applicability of mathematics, which is claimed to be contrary to the basic motive of Frege. This is because each of the number of different types required for each target.
In many cases, does not include the logic card. Is based on the theory of mathematical means to present a system capable of deriving the mathematical theory. This project starts from the set theory of Cantor. Cantor miele australia has made many achievements on eoteuna infinite, but did not present their systems to the axiom system (basic miele australia principles, did not indicate a strictly defined set of concepts.) Many paradoxes are presented as a system of set theory is a set of axioms miele australia arisen that need to be explicitly presented as: (must present the principles and conceptual reasoning that allows multiple mathematical theorem.) had Frege is planning to fail. The one that Zermelo perform these tasks. (Zermelo and the claims are similar to Hilbert.) miele australia
Learning Mathematics with the current standard, generally learn about the axiom system of set theory Zermelo-Fraenkel set theory was proposed. Most modern mathematicians mathematics knowledge, miele australia organize their ZFC (Zermelo-Fraenkel with axiom of choice) from that reasoning 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.)
Are presented in the theory ZFC through the FOL (First ordered logic). What we learned in Logic is for the logical system. ZFC is a theory about what special things (number theory is a theory about the number of either. Frege sees that this is reduced to the rules of logic system) should be introduced and basic theory principles (such as Newton's law is also axiomatic can be presented as FOL.) ZFC also use the basic rules of the FOL. The principle of ZFC is presented membership relation. The Frege defines this as the logical rule, but introduced x y Set theory is the principle. And introduces the concepts needed for this.
An empty set of elements, and, when x is element of y, {x} to be the set y as elements having.
Axiom 3: Subset axiom. Cause of Russell's miele australia paradox miele australia was Axiom V ( y x (x y Fx). The difference between axioms 3 and Axiom V is in x a., Where a refers to a specific set, ie it if there is any set is to give a limit of, so do not lay Axiom 3 contradictions, because its size is always some kind of constraint but Comprehension axiom.
Axiom 4: What is it that sets this power set of the set. If this and the use of Cantor miele australia Cantor's diagonal argument can be derived that in a kind infinity. We can not work describing the transcript P (ℕ) (because Uncountable) i.e., ℕ with 1: 1 correspondence is not. This is a set of technologies that we can not, Axiom 4 allows to introduce it into existence. Because this is a very powerful axiom Axiom of power set.
Axiom 6: The number of controversial miele australia axiom. There are several version, and the version is frequently used in mathematics John's Lemma. Zermelo, but fails to prove that every set can be well-ordering. As a proof of this because it implies miele australia the axiom 6. 6 axiom implies that the set is configured to select one element from a set of set of Disjoint.
zermelo n is the number of {..... { } .....} and {} are presented in the sense that the n. The foundation axiom is necessary because it is necessary to adopt in order to prevent miele australia recursive set such as x y y z z x. The easiest example of such a recursive set is x x. Foundation is also a condition of the well-ordering. This condition miele australia means that the starting point of the operation to configure the set yideonji any set. Therefore miele australia Foundation axiom that prevents the case when the dog is {} n, {} is infinite. It should be noted that the infinite set of infinite geotyiji miele australia the process for configuring a set of representations {} is not infinite.
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 Foundation miele australia axiom is x x accepts. . What this does not cause a problem of stress is because the constraint of the axiom 3 (this concept would not be present the concepts to be satisfied in all Fx {x: x x}. Is the element that all is set to.)
level 1: { }
level ω is the set that contains all the natural numbers as the sum of u again after applying the Power set operations. miele australia 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 Street type. In other words, from things around the level does not appear in the next level. miele australia 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, miele australia but the method is the same. 0 = , 1 = {0} = { }, 2 = {0,1} = { , { }} if presented in this way Frege n '= n {n} will be. Way of Frege and Russell is understood as normative. The reason for this is that it can reduce the size relationship between miele australia the natural number of defining in this way a natural number as membership relation. (According to the method of ZFC level due to Cumulative)
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