The "odd" or "counterintuitive" properties as you approach infinity, are a perfectly reasonable consequence of the general agreement to throw away information a

The "odd" or "counterintuitive" properties as you approach infinity, are a perfectly reasonable consequence of the general agreement to throw away information and treat infinity as all the same. But they're not all the same, they're simply large enough that any non-infinite numbers are below epsilon. Much like how infinitesimals are all equal to zero (below epsilon for any non-infinitesimal), but if they were all the same calculus wouldn't work.