Acceptance of transfinite numbers

In summary, transfinite numbers are numbers that are greater than any finite number, first introduced by Georg Cantor in the late 19th century. Their acceptance has led to advancements in mathematics, such as set theory and infinite series, and has resolved paradoxes and limitations in traditional mathematical thinking. They are represented using special notations and are still a topic of debate among mathematicians. Transfinite numbers have practical applications in fields such as computer science, physics, and economics.
  • #1
brianhurren
71
2
Are Transfinite numbers accepted as genuine numbers?
 
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  • #2
Of course they are. They are NOT part of the real number system but they have been defined, operations defined on them, properties deduced. Everything we require of numbers.

(Perhaps you should try to make clear exactly what you mean by numbers.)
 
  • #3
brianhurren said:
Are Transfinite numbers accepted as genuine numbers?

What does "genuine numbers" mean? How can a number system be not genuine?
 

What are transfinite numbers?

Transfinite numbers are numbers that are greater than any finite number, but are still considered to be a part of the number system. They were first introduced by the mathematician Georg Cantor in the late 19th century.

Why is the acceptance of transfinite numbers important?

The acceptance of transfinite numbers paved the way for new developments in mathematics, such as set theory and infinite series. It also helped to resolve paradoxes and limitations in traditional mathematical thinking.

How are transfinite numbers represented?

Transfinite numbers are represented using a special notation, such as aleph numbers (ℵ) or omega numbers (ω). These notations indicate the size or magnitude of the transfinite number.

Are transfinite numbers accepted by all mathematicians?

No, there is still some debate and skepticism surrounding the acceptance of transfinite numbers. Some mathematicians believe that they are purely abstract concepts and do not have a real existence, while others view them as useful tools for solving mathematical problems.

How are transfinite numbers used in real-world applications?

Transfinite numbers have been used in various fields such as computer science, physics, and economics. They have also been applied in game theory, cryptography, and other areas where infinite sets and non-standard analysis are relevant.

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