4 ms·

I don't really don't like the wording of the author - "infinity + 1" - because it isn't. The omegas are used to describe the ordering of the set and not, in any

I don't really don't like the wording of the author - "infinity + 1" - because it isn't. The omegas are used to describe the ordering of the set and not, in any way, the "magnitude" of the infinity. Well, I guess the catch phrase "infinity is a number - an infinite number of numbers" is just too good of a "deep wisdom" that people repeat it regardless of how much sense it makes. Anyway, omega + 2 tells us, to put it simply, that we have a countably infinite subset, followed by a finite subset of two elements. In the same way, omega + omega would mean one countably infinite subset followed by another. Note that this has no bearing whatsoever in the magnitude of the set. No matter how many omegas you add, you can still project the result onto the "basic" set of natural numbers. The labels don't matter, you aren't limited by the author's example of splitting even/odd. You can try it with sets {x, xx, xxx, ...} and {y, yy, yyy, ...} and they still project onto the naturals just fine.

mjw 15y ago
Nevertheless there is this useful notion of addition for ordinal numbers, as well as multiplication, exponentiation and so on. And so the 'omega + 1' notation is common. It is an ambiguous notation though -- you need to know whether it's ordinal arithmetic, or cardinal arithmetic, being used. Usually it's clear from context; and of course the two coincide on the plain old natural numbers. http://en.wikipedia.org/wiki/Ordinal_arithmetic http://en.wikipedia.org/wiki/Ordinal_arithmetic for anyone interested in investigating further. One interesting fact is that while ordinal arithmetic can satisfy a lot of the properties you'd expect, it's not commutative in general: omega+1 > 1+omega = omega. As for whether or not ordinals have the right to be considered 'magnitudes' in a sense, or whether only cardinals deserve that name. I think this stuff already strays so far from the realms of normal human intuition about magnitude that this would be more just an argument over definitions than anything else.
michaelochurch 15y ago
Cardinality is a measure of size, and (by definition) all countably infinite sets have the same number of elements based on the principle that any two sets for which we have a bijection between them are of the same size. Ordinals deal with something else: complexity of orderings upon sets. As the ordinals get "larger", the orderings they represent get increasingly complicated. Once we're past the Church-Kleene ordinal, they're so convoluted that we can't algorithmically compare them.

mjw 15y ago

Nevertheless there is this useful notion of addition for ordinal numbers, as well as multiplication, exponentiation and so on. And so the 'omega + 1' notation is common. It is an ambiguous notation though -- you need to know whether it's ordinal arithmetic, or cardinal arithmetic, being used. Usually it's clear from context; and of course the two coincide on the plain old natural numbers. http://en.wikipedia.org/wiki/Ordinal_arithmetic http://en.wikipedia.org/wiki/Ordinal_arithmetic for anyone interested in investigating further. One interesting fact is that while ordinal arithmetic can satisfy a lot of the properties you'd expect, it's not commutative in general: omega+1 > 1+omega = omega. As for whether or not ordinals have the right to be considered 'magnitudes' in a sense, or whether only cardinals deserve that name. I think this stuff already strays so far from the realms of normal human intuition about magnitude that this would be more just an argument over definitions than anything else.

michaelochurch 15y ago

Cardinality is a measure of size, and (by definition) all countably infinite sets have the same number of elements based on the principle that any two sets for which we have a bijection between them are of the same size. Ordinals deal with something else: complexity of orderings upon sets. As the ordinals get "larger", the orderings they represent get increasingly complicated. Once we're past the Church-Kleene ordinal, they're so convoluted that we can't algorithmically compare them.