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You would define it as a digit sequence, using ω + 1 as the indexing set. I would consider it to be most naturally an element of the hyperreal numbers, although
by klodolph 7y ago
You would define it as a digit sequence, using ω + 1 as the indexing set. I would consider it to be most naturally an element of the hyperreal numbers, although it is also contained in smaller extensions of the real numbers.
- hilbertseries 7y agoSure, you could define some sort of number system that has an infinite point digit. I doubt that’s what the original poster had in mind though.
- klodolph 7y agoThese number systems already exist, I’m not just making them up. If someone says 0.666…7 then you have a couple different ways you can take the discussion. You can say, “No! Real numbers don’t work like that!” or you can talk about what number systems would look like if you can do that. It turns out that there’s a lot to learn from the alternative number systems, including formulations of calculus without limits that are easier to understand from an intuitive perspective, yet equally rigorous. The field is called “nonstandard analysis”. It’s not taught in college.