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From finite to infinite: Do you know the true meaning of transfinite numbers?
In the world of mathematics, infinity is often portrayed as a fascinating subject. However, when it comes to "transfinite numbers", the depth and breadth of this concept often confuses many people. Transfinite numbers are those "infinite" numbers that are larger than all finite numbers. They include transfinite cardinals (numbers used to quantify the size of infinite sets) and transfinite ordinals (numbers used to represent infinite sets). sorted numbers). This article will explore these concepts in depth and give you a glimpse into the charm of transfinite numbers.
The term "transfinite" was first coined in 1895 by mathematician Georg Cantor, who wanted to avoid the ambiguous connotations of the word "infinity," although these numbers Not inherently finite.
By mathematical definition, any finite natural number can be used in at least two ways: as an ordinal number and as a cardinality. Cardinality is used to specify the size of a set, for example, "five marbles", while ordinal numbers are used to specify the position of a member of an ordered set, such as "third from the left" or "the first member of January". Twenty-seventh day". When these concepts are extended to transfinite numbers, there is no longer a one-to-one correspondence between the two. A transfinite cardinality describes the size of an infinite set, while a transfinite ordinal describes the position of a number in an ordered large set.
The most famous ordinals and cardinals among the transfinite integers are ω (Omega) and ℵ₀ (Aleph-null), which represent the starting point of infinity.
First, ω is the lowest transfinite ordinal, which is usually used to represent the ordinal type of natural numbers. ℵ₀ is the first transfinite cardinality, and it is also the cardinality of natural numbers. If the Axiom of Choice holds, then the next higher cardinality is ℵ₁. If this is not true, then there may be cardinalities that are larger than ℵ₁ but not equal to ℵ₀. It is worth noting that the continuum hypothesis proposes that there is no intermediate cardinality between ℵ₀ and the cardinality of the set of real numbers. This assumption cannot be proven in Zermelo–Frankel set theory, either by itself or by its negation.
Let's look at some concrete examples. In Cantor's ordinal number theory, every integer has its successor. The first infinite integer after all regular integers is named ω. More specifically, ω+1 is greater than ω, and ω·2, ω², and ω^ω are also larger numbers. In these contexts, arithmetic expressions involving ω specify an ordinal number that can be viewed as the set of all integers up to that number.
For the representation of infinite integers, the Cantor standard form provides a finite data sequence to represent it, but not all infinite integers can be represented using this standard form.
To complicate matters further, some infinite integers cannot be represented in Cantor form, and the first such integer is ω^(ω^(ω...)), called ε₀. This is a self-recursive number, where each solution ε₁, ..., εₖ, etc. makes the ordinal number larger. This process can be continued until a limit is reached, namely ε_(ε_(ε...)), which is the first solution of ε_α=α, meaning that when specifying all transfinite integers, an infinite name must be imagined. sequence.
In summary, the concept of transfinite numbers challenges our understanding of numbers and makes us think about the nature of infinity. It is not just the use of mathematical tools, but also involves deep philosophical thinking. We can't help but ask, when we face infinity, to what extent can the boundaries of our thinking reach?
