Yes, there are infinities of larger magnitude. It’s not a simple intuitive comparison though. One might think “well there are twice as many whole numbers as even whole numbers, so the set of whole numbers is larger.” In fact they are the same size.
Two most commonly used in mathematics are countably infinite and uncountably infinite. A set is countably infinite if we can establish a one to one correspondence between the set of natural numbers (counting numbers) and that set. Examples are all whole numbers (divide by 2 if the natural number is even, add 1, divide by 2, and multiply by -1 if it’s odd) and rational numbers (this is more involved, basically you can get 2 copies of the natural numbers, associate each pair (a,b) to a rational number a/b then draw a snaking line through all the numbers to establish a correspondence with the natural numbers).
Uncountably infinite sets are just that, uncountable. It’s impossible to devise a logical and consistent way of saying “this is the first number in the set, this is the second,…) and somehow counting every single number in the set. The main example that someone would know is the real numbers, which contain all rational numbers and all irrational numbers including numbers such as e, π, Φ etc. which are not rational numbers but can either be described as solutions to rational algebraic equations (“what are the solutions to “x^2 - 2 = 0”) or as the limits of rational sequences.
Interestingly, the rational numbers are a dense subset within the real numbers. There’s some mathsy mumbo jumbo behind this statement, but a simplistic (and insufficient) argument is: pick 2 real numbers, then there exists a rational number between those two numbers. Still, despite the fact that the rationals are infinite, and dense within the reals, if it was possible to somehow place all the real numbers on a huge dartboard where every molecule of the dartboard is a number, then throwing a dart there is a 0% chance to hit a rational number and a 100% chance to hit an irrational number. This relies on more sophisticated maths techniques for measuring sets, but essentially the rationals are like a layer of inconsequential dust covering the real line.
There are different things which could be called “infinite numbers.” The one discussed in the other reply is “cardinal numbers” or “cardinalities,” which are “the sizes of sets.” This is the one that’s typically meant when it’s claimed that “some infinities are bigger than others,” because e.g. the set of natural numbers is smaller (in the sense of cardinality) than the set of real numbers.
Ordinal numbers are another. Whereas cardinals extend the notion of “how many” to the infinite scale, ordinals extend the notion of “sequence.” Just like a natural number always has a successor, an ordinal does too. We bridge the gap to infinity by defining an ordinal as e.g. “the set of ordinals preceding it.” So {} is the first one, called 0, and {{}} is the next one (1), and so on. The set of all finite ordinals (natural numbers) {{}, {{}}, …} = {0, 1, 2, 3, …} is an ordinal too, the first infinite one, called omega. And now clearly {omega} = omega + 1 is next.
Hyperreal numbers extend the real numbers rather than just the naturals, and their definition is a little more contrived. You can think of it as “the real numbers plus an infinite number omega,” with reasonable definitions for addition and multiplication and such, so that e.g. 1/omega is an infinitesimal (greater than zero but smaller than any positive real number). In this context, omega + 1 or 2 * omega are greater than omega.
Surreal numbers are yet another, extending both the real and hyperreal numbers (so by default the answer is “yes” here too).
The extended real numbers are just “the real numbers plus two formal symbols, “infinity” and “negative infinity”.” This lacks the rich algebraic structure of the hyperreals, but can be used to simplify expressions involving limits of real numbers. For example, in the extended reals, “infinity plus one is infinity” is a shorthand for the fact that “if a_n is a series approaching infinity as n -> infinity, then (a_n + 1) approaches infinity as n -> infinity.” In this context, there are no “different kinds of infinity.”
The list goes on, but generally, yes-- most things that are reasonably called “infinite numbers” have a concept of “larger infinities.”
Is it possible for infinite numbers to be larger than others? Or are all infinite numbers equal?
Yes, there are infinities of larger magnitude. It’s not a simple intuitive comparison though. One might think “well there are twice as many whole numbers as even whole numbers, so the set of whole numbers is larger.” In fact they are the same size.
Two most commonly used in mathematics are countably infinite and uncountably infinite. A set is countably infinite if we can establish a one to one correspondence between the set of natural numbers (counting numbers) and that set. Examples are all whole numbers (divide by 2 if the natural number is even, add 1, divide by 2, and multiply by -1 if it’s odd) and rational numbers (this is more involved, basically you can get 2 copies of the natural numbers, associate each pair (a,b) to a rational number a/b then draw a snaking line through all the numbers to establish a correspondence with the natural numbers).
Uncountably infinite sets are just that, uncountable. It’s impossible to devise a logical and consistent way of saying “this is the first number in the set, this is the second,…) and somehow counting every single number in the set. The main example that someone would know is the real numbers, which contain all rational numbers and all irrational numbers including numbers such as e, π, Φ etc. which are not rational numbers but can either be described as solutions to rational algebraic equations (“what are the solutions to “x^2 - 2 = 0”) or as the limits of rational sequences.
Interestingly, the rational numbers are a dense subset within the real numbers. There’s some mathsy mumbo jumbo behind this statement, but a simplistic (and insufficient) argument is: pick 2 real numbers, then there exists a rational number between those two numbers. Still, despite the fact that the rationals are infinite, and dense within the reals, if it was possible to somehow place all the real numbers on a huge dartboard where every molecule of the dartboard is a number, then throwing a dart there is a 0% chance to hit a rational number and a 100% chance to hit an irrational number. This relies on more sophisticated maths techniques for measuring sets, but essentially the rationals are like a layer of inconsequential dust covering the real line.
There are different things which could be called “infinite numbers.” The one discussed in the other reply is “cardinal numbers” or “cardinalities,” which are “the sizes of sets.” This is the one that’s typically meant when it’s claimed that “some infinities are bigger than others,” because e.g. the set of natural numbers is smaller (in the sense of cardinality) than the set of real numbers.
Ordinal numbers are another. Whereas cardinals extend the notion of “how many” to the infinite scale, ordinals extend the notion of “sequence.” Just like a natural number always has a successor, an ordinal does too. We bridge the gap to infinity by defining an ordinal as e.g. “the set of ordinals preceding it.” So {} is the first one, called 0, and {{}} is the next one (1), and so on. The set of all finite ordinals (natural numbers) {{}, {{}}, …} = {0, 1, 2, 3, …} is an ordinal too, the first infinite one, called omega. And now clearly {omega} = omega + 1 is next.
Hyperreal numbers extend the real numbers rather than just the naturals, and their definition is a little more contrived. You can think of it as “the real numbers plus an infinite number omega,” with reasonable definitions for addition and multiplication and such, so that e.g. 1/omega is an infinitesimal (greater than zero but smaller than any positive real number). In this context, omega + 1 or 2 * omega are greater than omega.
Surreal numbers are yet another, extending both the real and hyperreal numbers (so by default the answer is “yes” here too).
The extended real numbers are just “the real numbers plus two formal symbols, “infinity” and “negative infinity”.” This lacks the rich algebraic structure of the hyperreals, but can be used to simplify expressions involving limits of real numbers. For example, in the extended reals, “infinity plus one is infinity” is a shorthand for the fact that “if a_n is a series approaching infinity as n -> infinity, then (a_n + 1) approaches infinity as n -> infinity.” In this context, there are no “different kinds of infinity.”
The list goes on, but generally, yes-- most things that are reasonably called “infinite numbers” have a concept of “larger infinities.”