Presumably you are aware of the host of problems introduced to mapping reality when concepts of infinity are included? There is literally no problem on earth in which the concept of infinity is needed for a solution. In the overwhelming majority of problems, a concept of infinity is either not needed or all, or leads to wrong answers when employed. The planet is finite, the mind is finite, the observed universe is finite. Even infinitesimals are make believe, I don’t think we have ever observed anything less than 10^-20 m, and if I am wrong make it 10^-50 m to put a wall around it.
Whether you think there are some places where infinity helps or not, would you disagree that there are a vast ream of useful things to be done with numbers where infinity would never be missed?
So why toss out the beginnings of a system that doesn’t include something of such limited utility as a detailed property of a set of concepts that aren’t even needed to accomplish everything technical in the world that has to date been accomplished?
I think the most important reason for keeping infinity around is that the simplest system which describes how numbers work implies the existence of infinitely many numbers. It is much easier to work in this system and then ask “is my answer smaller than 10^50?” than to work in a system using only numbers less than 10^50 and ask “does the result I got actually exist?” at every step.
Similarly, the simplest system that can describe lengths of things is the real numbers. In a way these are make-believe, but it would be ridiculous to try to describe the length of a diagonal of a square without having the concept of the square root of 2.
Comparing the different sizes of infinity is, in its simplest case, a description of how these two systems interact. Don’t tell me that’s not useful.
Presumably you are aware of the host of problems introduced to mapping reality when concepts of infinity are included? There is literally no problem on earth in which the concept of infinity is needed for a solution. In the overwhelming majority of problems, a concept of infinity is either not needed or all, or leads to wrong answers when employed. The planet is finite, the mind is finite, the observed universe is finite. Even infinitesimals are make believe, I don’t think we have ever observed anything less than 10^-20 m, and if I am wrong make it 10^-50 m to put a wall around it.
Whether you think there are some places where infinity helps or not, would you disagree that there are a vast ream of useful things to be done with numbers where infinity would never be missed?
So why toss out the beginnings of a system that doesn’t include something of such limited utility as a detailed property of a set of concepts that aren’t even needed to accomplish everything technical in the world that has to date been accomplished?
I think the most important reason for keeping infinity around is that the simplest system which describes how numbers work implies the existence of infinitely many numbers. It is much easier to work in this system and then ask “is my answer smaller than 10^50?” than to work in a system using only numbers less than 10^50 and ask “does the result I got actually exist?” at every step.
Similarly, the simplest system that can describe lengths of things is the real numbers. In a way these are make-believe, but it would be ridiculous to try to describe the length of a diagonal of a square without having the concept of the square root of 2.
Comparing the different sizes of infinity is, in its simplest case, a description of how these two systems interact. Don’t tell me that’s not useful.