Uncountability is an idea that skeptics/rationalists/less-wrongers should treat with a grain of salt.
Inside of a particular system, you might add an additional symbol, “i”, with the assumed definitional property that “i times i plus one equals zero”. This is how we make complex numbers. We might similarly add “infinity”, with an assumed definitional property that “infinity plus one equals infinity”. Depending on the laws that we’ve already assumed, this might form a contradiction or cause everything to collapse, but in some situations it’s a reasonable thing to do. This new symbol, “infinity” can be definitionally large in magnitude without being large in information content—bits or bytes required to communicate it.
A specific generic member of an uncountable set is infinitely large in information content—that means it would take infinite time to communicate at finite bandwidth, infinite volume to store at finite information density. It is utterly impractical.
It is reasonable (because it is true) to believe that every mathematical object that a mathematician has ever manipulated was of finite information content. The set of all objects of finite information content is merely countable, not uncountable.
What I’m trying to say is: Don’t worry about uncountability. It isn’t physical and it isn’t practical. Mathematicians may be in love with it (Hilbert’s famous quote: “No one shall expel us from the Paradise that Cantor has created.”), but the concept is a bit like that “infinite” symbol—infinite in magnitude, not in information content.
I strongly recommend that, if you haven’t already, you learn enough introductory calculus to understand what it means to take the limit of an expression as a variable approaches infinity. You are making a common mistake here by conflating your intuitive understanding about infinity with its meaning in stricter mathematical contexts.
Uncountability is an idea that skeptics/rationalists/less-wrongers should treat with a grain of salt.
Inside of a particular system, you might add an additional symbol, “i”, with the assumed definitional property that “i times i plus one equals zero”. This is how we make complex numbers. We might similarly add “infinity”, with an assumed definitional property that “infinity plus one equals infinity”. Depending on the laws that we’ve already assumed, this might form a contradiction or cause everything to collapse, but in some situations it’s a reasonable thing to do. This new symbol, “infinity” can be definitionally large in magnitude without being large in information content—bits or bytes required to communicate it.
A specific generic member of an uncountable set is infinitely large in information content—that means it would take infinite time to communicate at finite bandwidth, infinite volume to store at finite information density. It is utterly impractical.
It is reasonable (because it is true) to believe that every mathematical object that a mathematician has ever manipulated was of finite information content. The set of all objects of finite information content is merely countable, not uncountable.
What I’m trying to say is: Don’t worry about uncountability. It isn’t physical and it isn’t practical. Mathematicians may be in love with it (Hilbert’s famous quote: “No one shall expel us from the Paradise that Cantor has created.”), but the concept is a bit like that “infinite” symbol—infinite in magnitude, not in information content.
I strongly recommend that, if you haven’t already, you learn enough introductory calculus to understand what it means to take the limit of an expression as a variable approaches infinity. You are making a common mistake here by conflating your intuitive understanding about infinity with its meaning in stricter mathematical contexts.