Is infinity bigger than infinity+1?
In: 1
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.
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