I think mathematicians have got it wrong with infinity. Infinity is infinity. One infinity cannot be larger than another infinity. What do you think?
Actually, you do kind of have a point, despite what others here and most mathematicians have thought for a long time.
Infinity is not a number. It does not have a size. It is a concept that indicates *potentiality*. The potentiality of what? To progress in a sequence.
Let us take the counting numbers. Let us say that you start listing them. Well, you obviously have to stop sometime, you cannot keep doing it forever. But, no matter how many of them you list, you never “run out” of counting numbers. There are always more of them you could have listed had you kept going.
Or, take the digits of “pi”. You can keep listing those digits for a long time, but you have to stop at some point. But, no matter at which point you *do* stop, there are always further digits you might have listed.
This is what infinity indicates: The fact that there are certain mathematical sequences which you can progress along until a certain point. But, at no matter which point you do stop, there is always the potential to have progressed further.
It indicates that the progression along any sequence must be finite, but that no matter where you stop progressing, you do not run of terms and that if you had kept going, you could always have identified further terms.
Infinity simply indicates the potential to progress in a sequence of mathematical steps, no matter where one does actually stop. There is always the *potential* to have continued.
Alright, so if infinity indicates merely a potential, then where does size enter into it?
It doesn’t. It makes no sense to discuss the “size” of a potential such as this. Either the potential exists or it does not. There are no “potentials” that are larger than other “potentials”.
What then to make of the claims that some “infinite” sets are larger than others? Well, not that some infinities are larger than others.
This might surprise you: infinite sets do not have sizes! They are simply infinite. To say that they have a size implies that you can count and quantify all of their elements. But, you can’t, that is what it means to be an infinite set!
So, if infinite sets have no size, then there is no basis for comparing their sizes!
It would look much the same as it does now, ignoring parts of Earth (and possibly other worlds with intelligent life) that have been changed by people.
Mathematics is a science of method invented by people to help them measure things. Without it, we would lack the ability to do much science and we would know almost nothing about the world or our universe. Without it, we would know nothing about engineering and we would be unable to build most of the technology we have.
But, other than the fact that the universe would lack all of those things that people have built, it would be much the same.
It is not as though mathematics is part of the universe or as though it needs mathematics to work the way it does. Despite what many physicists believe, mathematics is not fundamental to the universe. It is just something people use to measure things in the universe.
I know that people cannot understand the “unreasonable success” of mathematics. It is not unreasonable at all, it is entirely predictable and obvious, if you understand what mathematics is: a method of quantifying relationships and performing measurements!
If you know that, then why should it be surprising that mathematics is able to … quantify and measure the universe?
It is a science of method. Methods of quantifying relationships, quite often for the purpose of performing measurements of physical things.
It is NOT the same thing as the quantities being measured. The relationships and properties being measured are discovered.
Mathematics, is a science of method and those methods must be invented, created. They are no more “discovered” than any other such methods are “discovered”.
One does not “discover” the process of programming computers, one invents those processes. One does not “discover” the process of measuring circles. One invents processes for doing so. Unless of course, they already exist and one discovers what someone else did.
Even numbers are not “discovered”. They are invented as an important conceptual method which forms a basis for mathematics.