IIRC Depends if you talk about cardinal or ordinal numbers.
What I remember:
In cardinal numbers (the normal numbers we think of, which denote quantity, etc.) have their maximum in infinity.
But in ordinal numbers (which denote order - first, second, etc.) Can go past infinity - the first after infinity is omega. Then omega +1. And then some bigger stuff, which I don’t remember much, like aleph 0 and more.
No cardinal and ordinal numbers continue past the “first” infinity in modern math. I.e. The cardinal number denoting the cardinality of the natural numbers (aleph_0) is smaller than the one of the reals.
Edit: In modern systems aleph_0 = omega btw. Omega denotes ordinal and aleph denotes cardinals.
So wait, you can’t have numbers larger than infinity, but you can order them “past infinity”? I’m trying to wrap my head around the concept, and the clearest thing I can get at the moment is that the "infinity+1"th number is infinity… would that be right?
No you can have numbers past infinity op is wrong.
As for how to order past the first infinity it’s easy.
Of course first you have 1 < 2 < 3 < 4 < … Then you take a new number not equal to any of the others let’s call it omega. Define omega to be larger than the others. So 1 < omega, 2 < omega,…
This you can of course continue even further by introducing omega + 1 which is larger than omega and therefore larger than all natural numbers.
You can continue this even further by introducing a new number let’s call it lambda that is bigger than all omega + x where x is a natural number.
This can be continued forever i.e. an infinite amount of times.
Edit: that is meant by ordinal number as you define a unique order each step.
The problem is that the concept of cardinality and ordinality is the same in the finite case. That is numbers that tell you how many things there are can also be used to sort them.
This does not work past the first infinity. If you add omega to the natural numbers then the amount of numbers you have is still the first infinity.
But there are bigger cardinal infinities than the first one. For example the cardinality of the real numbers. I.e. There are more real numbers than natural numbers.
IIRC Depends if you talk about cardinal or ordinal numbers. What I remember: In cardinal numbers (the normal numbers we think of, which denote quantity, etc.) have their maximum in infinity. But in ordinal numbers (which denote order - first, second, etc.) Can go past infinity - the first after infinity is omega. Then omega +1. And then some bigger stuff, which I don’t remember much, like aleph 0 and more.
No cardinal and ordinal numbers continue past the “first” infinity in modern math. I.e. The cardinal number denoting the cardinality of the natural numbers (aleph_0) is smaller than the one of the reals.
Edit: In modern systems aleph_0 = omega btw. Omega denotes ordinal and aleph denotes cardinals.
So wait, you can’t have numbers larger than infinity, but you can order them “past infinity”? I’m trying to wrap my head around the concept, and the clearest thing I can get at the moment is that the "infinity+1"th number is infinity… would that be right?
No you can have numbers past infinity op is wrong.
As for how to order past the first infinity it’s easy.
Of course first you have 1 < 2 < 3 < 4 < … Then you take a new number not equal to any of the others let’s call it omega. Define omega to be larger than the others. So 1 < omega, 2 < omega,…
This you can of course continue even further by introducing omega + 1 which is larger than omega and therefore larger than all natural numbers.
You can continue this even further by introducing a new number let’s call it lambda that is bigger than all omega + x where x is a natural number.
This can be continued forever i.e. an infinite amount of times.
Edit: that is meant by ordinal number as you define a unique order each step.
The problem is that the concept of cardinality and ordinality is the same in the finite case. That is numbers that tell you how many things there are can also be used to sort them.
This does not work past the first infinity. If you add omega to the natural numbers then the amount of numbers you have is still the first infinity.
But there are bigger cardinal infinities than the first one. For example the cardinality of the real numbers. I.e. There are more real numbers than natural numbers.