Is infinity bigger than infinity+1?
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Infinity + 1 doesn’t really make sense because infinity isn’t an actual value, its a concept.
It has been shown that there are larger and smaller infinities though.
Basically if you have an infinite set of values from 0 to 1 in decimal form, you have an infinite list of numbers. But that infinite list is smaller than an infinite list of all decimals from 0 to 10, because the list from 0 to 10 contains all the values from 0 to 1 and more, so its larger.
There are different types of infinity with different sizes but Infinity +1 is the same size as infinity.
It comes down to being able to map the different sets of numbers onto each other. Adding plus 1 is trivial to map to existing infinite set.
You get into stuff like aleph numbers and things like that if you look for large infirmities, but all the countable stuff like integers, or rational numbers is all the same size. Adding a single element won’t make any of them bigger than the others. In fact adding infinitely many of them won’t make them bigger either.
No, partly because infinity is the concept of never ending series of points, not a number we can do arithmatic to.
However, we can define sets to be infinite in size, and then we can do some mathematical things to those sets, but they’re not numbers.
One of the things we can sometimes do to an infinite set is “list” the points on that set. We can take the example of the positive whole numbers. 1, 2, 3… You can list them out in that order.
Now if you want to show if a different infinite set is the same size, all you have to do is match the points from your new set onto the points from this first one. So let’s say you want to do “infinity (meaning the positive whole number list) + 1
Well, we can do that, just add 1 to all of your points.
1+1=2, 2+1=3, 3+1=4…
But now we can take that second list, and match it 1:1 to the first list.
1 matches with 2, 2 matches with 3, 3 matches with 4, and so on. And because there is no last positive whole number, it all matches. And you have proven that the list of positive whole numbers is equal to the positive whole numbers + 1.
The idea of comparing sizes of infinity just doesn’t really work. The question doesn’t have a satisfying answer and “infinity plus one” doesn’t really mean anything.
Infinity is not a number so you can’t just add things to it like any other number.
Infinity is weird and if anything about it seems to make sense you’ve probably misunderstood it. If it seems weird and confusing and contradictory then you’re probably on the right track.
Infinity is a concept, not a number.
There are several kinds (and several values of most of these kinds) of infinity in math. Some of which are numbers (ordinals like ω or ε_0, cardinals like ℵ_0), some of which are not (∞).
However, I don’t think there are any of these where infinity > infinity + 1. The two might be equal though
Infinity is a concept that has a lot of ways to be defined properly. There is the easy solution of saying that “infinity+1 doesn’t exists”, but with a good number of definitions of infinity we can actually define “infinity+1”.
So, how does “Infinity+1” works? First let’s agree on which infinite number we select here. Infinite numbers are called **ordinals**. In our case, we take what is called the **countable** infinity. That’s the one you obtain when you do 1, 2, 3, etc, “Infinity”.
* “Infinity+1” is of exactly the **same size** as “Infinity” (or of the **same cardinality**, if you want to use the dedicated mathematical term). There are bigger ordinals than that. in particular the **continuous infinity** is bigger than both “Infinity” and “Infinity+1”, but that’s another subject.
* “Infinity+1” as in “Infinity and we add an element at the end” is a **greater** ordinal than “Infinity”. In some ways, you can say that “Infinity+1” is **bigger by a negligible amount** than “Infinity”, which mean that if you look at how they are ordered, “Infinity+1” comes after “Infinity”, but if you look at their size then they both have the same size.
No.
I find this best to explain with a visual metaphor.
Lets use a infinite set as our first infinite: the set of numbers greater than or equal to 1. If we plot this on a number line, the ray would start at 1 and stretch out to infinity.
For an “infinity+1”, lets use the set of numbers greater or equal to 0. If plotted on the number line, the ray would start at 0. However, if we apply a simple linear transformation to the other infinity: shift it’s starting point from 1 to 0, they become the same ray, so they are the same size.
This is true of any transformation, no matter how complex the transformation. For example, we know that the set of all numbers less than or equal to zero is the same size as the original infinity as it’s just the same ray flipped in the opposite direction. However, we know that the set of all positive and negative numbers is large than our original infinity because it stretches out to infinity in both directions. No matter how we flip, stretch, or move our original infinity it only goes to infinity in one direction, so we can’t transform it into the other infinity.
There are different kinds of infinity. But infinity+1 is always the same size as infinity.
For example, there are an infinite number of even numbers, and an infinite number of all numbers.
Imagine taking each even number and matching it with a different natural number. 2 matches to 1, 4 to 2, 6 to 3, 8 to 4, and so on.
Now, we can never run out of numbers. We can always add 1 or 2 to make a new number. And that new number would also have a match.
Since this is the case, infinity+1 is the same size as infinity.
A scenario where two infinities aren’t the same size would be as follows…
Imagine the infinity of all natural numbers, N. And the infinity of all real numbers, R.
R is larger than N.
This is because, for real numbers, there are an infinite number of subdivisions between any two real numbers.
While not strictly true, the size of R is about the size of N squared. It’s an infinity compromised of smaller infinities.
A hotel with infinite rooms which are all full can fit 1 person in if they turn up…
And then a coach with an infinite amout of people turn up, they all fit in…
And then an infinte amount of coaches…
https://en.wikipedia.org/wiki/Hilbert’s_paradox_of_the_Grand_Hotel?wprov=sfti1
The question doesn’t make sense. Now, if the question was: is infinity+1 larger than infinity, at least there’s some logical progression there. The more important question here should be, why are you asking this stupid shit and wasting everyone’s time here?
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