Though, I'd say that ordering is something that can only be defined using our sense-perception notion of counting discrete objects intuitively. For example, if you have 1 candy on a table, and then add a second candy on the table, then you have more candies because you can touch, feel, see more candies on the table, and therefore 2 > 1. This video takes this intuitive notion and places it inside set theory, much like the von neumann definition of natural numbers using only empty sets (cups stacked in cups). I haven't seen any work by anyone anywhere that uses pure logic alone to deduce the order of natural numbers without appealing to sense-perception, which, honestly, might be a big mistake for mathematics. It might be part of why quantum phenomena seems so unusual to us right now, but that's the hunch that I get; perhaps things/the true nature of mathematics *is not, in fact, discrete* and perhaps we've got it all wrong in thinking that natural numbers are "God Given." I'm starting to think that it is actually the real and complex numbers which are truly "God Given," and that our insistence on relying on discrete numerals is fundamentally flawed.
Order relations are well-defined in set theory, as is the von Neumann universe, including the ordering of the natural numbers. There's no need to rely on intuitive sense perception in order for any of it to be logically coherent and indeed intuitive.
“Ordering them in a natural way”
I see what you did there
Though, I'd say that ordering is something that can only be defined using our sense-perception notion of counting discrete objects intuitively. For example, if you have 1 candy on a table, and then add a second candy on the table, then you have more candies because you can touch, feel, see more candies on the table, and therefore 2 > 1.
This video takes this intuitive notion and places it inside set theory, much like the von neumann definition of natural numbers using only empty sets (cups stacked in cups). I haven't seen any work by anyone anywhere that uses pure logic alone to deduce the order of natural numbers without appealing to sense-perception, which, honestly, might be a big mistake for mathematics. It might be part of why quantum phenomena seems so unusual to us right now, but that's the hunch that I get; perhaps things/the true nature of mathematics *is not, in fact, discrete* and perhaps we've got it all wrong in thinking that natural numbers are "God Given." I'm starting to think that it is actually the real and complex numbers which are truly "God Given," and that our insistence on relying on discrete numerals is fundamentally flawed.
Order relations are well-defined in set theory, as is the von Neumann universe, including the ordering of the natural numbers. There's no need to rely on intuitive sense perception in order for any of it to be logically coherent and indeed intuitive.
This video goes by too fast....