Questions about the foundations of mathematics are primarily questions about set theory

The historical cognitive process is just the opposite, the ancient Egyptian people used arithmetic at will, and after Newton invented calculus, the differentiation and integral operation were also arbitrary, and the existence and continuity of the function limit were not considered at all, and later due to the needs of the development of the discipline and too many loopholes in the original theory, people began to gradually logic and axiomatization, first Weylstrass used -δ language to define the limit, and then used the rational number sequence to define the real number, and after the real number problem was solved, he began to consider how to define the rational number and even the whole number and the natural number, In the same way, Piano made the axiom of natural numbers, after Cantor's set theory came out, all this had to be redefined from set theory, after Russell's paradox appeared, he made axiom set theory zfc, and finally by the Bourbaki school to make it what it is now, here, first define the potential and ordinal numbers of the set, and then derive the Piano axiom of natural numbers from the set axioms, and then derive the addition and multiplication of natural numbers from Piano's axioms, and then use logical equivalence classes to derive integer subtraction and rational number division, Later, the limit operation of real numbers is derived from the sequence of rational numbers, and the later differential, integral, series, etc. are all based on the limit operation, Bourbaki summarized all mathematics into three basic structures: algebraic structure, order structure, topological structure, you can look at "Ancient and Modern Mathematical Thought (1 4 volumes)> or similar works on the history of the development of modern mathematics, mathematics and its understanding" (Gao Longchang) to introduce mathematical ideas to graduate students, very interesting.

Sets are the original concept, and so is 1+1=2.

There are also two straight lines in the plane that never intersect in parallel.

They're all axioms, and they're recognized as being the roots of a tree in mathematics.

You can see [Tao Zhexuan's actual analysis] that there are detailed answers in chapters 1 to 4.

Yes, because the meaning of can be less than or equal to.

2 3 3 3 3 are all correct.

So a+1 x where a+1 can be equal to x x x can also be equal to 2a-1, so a+1 can = 2a-1

Yes, a+1=2a-1, a=2, a+1=2a-1=3, and there is only one element in the set that is a=

There are a total of 8 + 10 + 9 = 27 non-conforming records.

Seven of the two items were unqualified, and 14 records were removed.

1 type of three unqualified, remove 3 records.

The rest is a non-conforming record, 27-14-3 = 10, 10 unqualified records, that is, 10 kinds of unqualified products for a total of 52 products, remove two unqualified products, three unqualified products, one unqualified product, and the rest is three fully qualified products, 52-7-1-10 = 34

A: 5 people who participated in the astronomy and literature groups at the same time.

B: 5 people who participated in both literature and physics groups.

C: 3 people who participated in the group of physics and astronomy at the same time.

a, b, and c all include (implicitly) those who participate in three hobby groups at the same time——— a: low temperature failure, solubble content not up to standard.

B: Low temperature is unqualified, and the seam cutting performance is unqualified.

C: The soluble content is not up to standard, and the seam cutting performance is unqualified.

According to the title, it can be known that a+b+c=7, but a, b, and c do not include three unqualified products at the same time.

Therefore, example 2 can be done with the formula of the repulsion principle, and if you want to use the principle of repulsion in example 1, you need to get 8+10+9-7-(3x the number of unqualified products in all three categories) +1=8+10+9-7-3x1+1=18

It's different.,The first question is a qualified supplement.,The second question is not a supplement.。

The symbol n(a) indicates the number of elements in the set a.

Able to set Chinese chess set A, Go set B, chess set C.

Known: n(a)=20, n(b)=19, n(c)=18, n(a b)=7, n(a c)=8, n(b c)=5, n(ab c)=3.

n(a∪b∪c)

n(a)+n(b)+n(c)-n(a∩b)-n(b∩c)-n(a∩c)+2*n(a∩b∩c)

43 There are 50-43 = 7 players who can't play all three kinds of chess.

Solving the system of equations on the right gives x=2,y=1, so it should be expressed as (2,1), and the title is expressed as (1,2), which of course does not belong to the set.

Hope, thank you.

"Set Theory" is a product of formal logic thinking, and under the "Set Theory" of form and god logic, there is also a "Body Theory". "Set Theory" is a general theory that individuals form collectives, while "Body Theory" is a general theory that parts constitute individuals. Together, these two treatises become two indispensable parts of the Theory of Objects.

A basic sub-discipline of mathematics that studies general sets. Set theory occupies a unique place in mathematics, and its basic concepts have permeated all areas of mathematics. Set theory or set theory is a mathematical theory that studies sets (a whole made up of a bunch of abstract objects) and encompasses the most basic mathematical concepts such as sets, elements, and member relationships.

In most formulations of modern mathematics, set theory provides the language in which mathematical objects are to be described. Set theory and logic, together with first-order logic, form the axiomatic basis of mathematics, and the undefined terms "set" and "set member" are used to formally construct mathematical objects.

In naïve set theory, a set is seen as a self-evident concept such as a whole made up of a collection of objects.

In axiomatic set theory, sets and set members are not defined directly, but rather some axioms that can describe their properties are first defined. Under this idea, sets and set members are like points and lines in Euclidean geometry, and are not directly defined.

Chinese name: set theory.

Main articles: Sets (mathematics) and Set algebra.

Features: In Euclidean geometry and not directly defined.

Significance: It is the foundation of the whole of modern mathematics.

Historical role. FunctionAccording to the modern mathematical viewpoint, the object of study of each branch of mathematics is either a set with a specific structure, such as a group, a ring, or a topological space, or it can be defined by a set (such as natural numbers, real numbers, functions). In this sense, set theory can be said to be the foundation of modern mathematics as a whole.

It solves the learning difficulty of high school students.

The equipotential principle of set theory was prepared by Kanghe Yuto to construct a theoretical and logical basis for modern analysis, not to describe the "world of common sense". It is absurd to try to use "common sense" to refute the principle of equipotentiality. It's like Zheng Chun thinks about the infinite in real life, because you can only give examples of the infinite potential (for example, the repetition between practice and cognition until infinity) in the case of truth, but you can't give examples of the infinite reality.

As long as it can form a logically consistent system, it is the correct foundation under the modern analytical system.

As a constructive principle, Cantor's theoretical assumptions can be replaced, as has been articulated in the study of the axioms of controversy. However, if a new system can be formed by replacing some axioms, it can only describe the new system, and cannot describe the original system as "wrong".