> Here's how: consider the set of numbers that cannot be described using any finite collection of symbols. Such numbers must exist because there are only a countably infinite number of numbers that can be described using a finite collection of symbols, but there are an uncountably infinite number of real numbers.
But there are uncountably many finite collections of symbols. Sorry, you will have to take the axiom of choice out of my dead cold hands. Banach-Tarski just tells us if we use un-measurable sets, they don't have much to do with our understanding of physical space. But that's fine, because sets are just collections, and physical space is undoubtedly more than that.
But there are uncountably many finite collections of symbols. Sorry, you will have to take the axiom of choice out of my dead cold hands. Banach-Tarski just tells us if we use un-measurable sets, they don't have much to do with our understanding of physical space. But that's fine, because sets are just collections, and physical space is undoubtedly more than that.