By Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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Can the set N represent any number at all? We can never finish writing the list of its elements. But the mere fact that we can’t write the whole list doesn’t mean that the set itself does not exist. We can imagine it, so in the mathematical world, it exists! Mathematicians define the number represented by the entire set N as a form of “infinity” and denote it using the last letter in the Greek alphabet, omega, in lowercase (ω). ” In formal terms, ω is called an infinite ordinal or transfinite ordinal, and it has some strange properties.

Click here for terms of use. 36 Natural Numbers and Integers Start here and go on forever Figure 3-1 The number 0 can be defined as the null set. We can show how it starts to generate natural numbers by placing it at the beginning of an endless string of points. Building new numbers Let’s make up a rule that we can use to generate new natural numbers, one after another. Suppose we’ve built a certain natural number. Call it n. If we want to create the next higher natural number, n + 1, we can take all the natural numbers up to and including n, make them into elements of a set, and then call that set the new number.

In theory, those elements could be anything, such as apples, stars, atoms, or people. But it’s convenient to use all the natural numbers less than p as those elements. That allows us to build the set of natural numbers, one on top of the other, like stacking coins. Just as each coin in the stack rests on all the coins below itself, every element p in the set of natural numbers “rests on” all the natural numbers “below” itself. Once this process is defined, it sets in motion a mathematical “chain reaction” that never ends.

### Algebra Some Current Trends by Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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