Here’s one way to prove my point: line up all the natural numbers going forever in one direction. Then side by side with it line up every multiple of ten. Both lines go on forever, but one is part of the other. YOU guys are taking the smaller set, taking away the gaps, and putting it side by side with the set of natural numbers, starting at one. But it is philosophically impossible to pull that line back like that! Imagine you are holding two ropes, both of which are going forward to infinity. It is not possible to pull one of them behind your back.
Proven: all odd numbers are not equal to all natural numbers
I mean this as charitable and graciously as possible. You are literally, not figuratively, talking nonsense. And the only thing the above proves is you don’t understand the concept of infinity. Or the basics of sets, perhaps. The set of natural numbers, and the set of multiples of ten are different sets that share elements. They’re not part of one another, except in perhaps a descriptive sense. If both sets are without limits, then there is no ‘smaller set’ to take gaps away from. Nor are there ‘gaps’ at all. The difference in elements aren’t ‘gaps’, they’re just different elements. The set of all natural numbers will always have another element. We’ll call this set George. So too will the set of every number divisible by one billion. We’ll call this set Larry. So let’s put the two sets together in a race, shall we? And see what happens.
Each set element is the same as one lap of a race track. George’s set is {1, 2, 3, 4, 5…N+1) So, in ten elements it will have gone 10.
Larry’s set is the element in George times one billion. So, in those same ten elements, Larry will also have gone 10. And for every element that George adds, Larry adds one too. George will NEVER have more elements than Larry. They will always have the exact same elements, when the sets are unbounded. These extra elements that George has are irrelevant to Larry - these elements aren’t in the set we’ve defined before even listing the elements.
You WOULD be correct if the sets are bounded. If we only cared about the first ten-billion elements, then yes. George will have more elements than Larry. But the sets, by definition, are infinite and without bounds. We’re not talking about distances, or weights, time spent, or counting on a number line. We’re just talking about elements in a set. 2, 3, 4, etc. aren’t elements Larry can have due to the definition of Larry. So they don’t matter. Every element George has will have a corresponding element in Larry. It’ll be like a ladder, where the rungs bridge the elements of the sets that are the side-parts. Every rung will connect an element of George to an element of Larry. Forever. There’ll be no holes, or gaps. There will always be a one-to-one correspondence between the elements of both sets. The sums of both sets will be infinite. The number of elements in both sets will be infinite. I don’t know how many more analogies I can spin to describe it.
