schapel wrote:Maybe this discussion is about something that confused some of my classmates in computer science: there are an infinite number of natural numbers, but all natural numbers are finite.
If you always get a natural number as a result of adding two natural numbers, then there is an endless supply of natural numbers. If I produce a natural number and claim it is the largest, you can always add 1 and prove me wrong. Therefore, there are an infinite number of natural numbers. However, each natural number can be written down in decimal form. There is no natural number that represents infinity.
If you have a set of natural numbers, it's either the case that:
1) Adding two natural numbers results in an element which is not in the set, meaning that the set is not closed with respect to addition, or
2) If the set is closed with respect to addition, there are an infinite number of natural numbers.
However, there is no natural number that represents infinity, so there is no such idea as infinity + 1 in this set with the + operator on natural numbers, because infinity is not a natural number.
I suppose the confusing concept is that the number of elements in the set of natural numbers cannot be expressed in terms of a natural number. If it were possible to do so, that number would be the largest element of the set and addition would not be closed over the natural numbers.
And thus the whole notion of "ordinal arithmetic" was born.
In ordinal arithmetic, you start with the same good old natural numbers: 0,1,2,3... . But you also have a rule which states that "every series of consecutive 'ordinal numbers' starting with 0, is in itself a an ordinal number.
So in the ordinal world, the infinite series 0,1,2,3.... is a number. Not a natural number, to sure, but an ordinal number named ω.
Of-course, once you have ω, there's nothing to stop you from continuing: ω+1, ω+2, ω+3...
Now, there's still no "largest ordinal". But every sequence of ordinals, finite or infinite, can be represented by an ordinal (unlike the set of natural numbers). This does come at a price (for one, the class of all ordinals is too large to be considered a "set"), but ordinal arithmetic does have its appeal (and its uses). It also retains one important feature of the natural numbers: for any ordinal k, k+1>k.