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I think it's just we don't have a way to say/write these numbers. So you make up a way to write them (-1 and 1) and continue. The numbers don't matter and you
by dubcanada 14d ago
I think it's just we don't have a way to say/write these numbers. So you make up a way to write them (-1 and 1) and continue.
The numbers don't matter and you could replace -1 and 1 with anything. It's just easier to begin your new fake number at - 1 and 1. Because position does matter.
- xg15 14d agoThere has to be some procedure how to come up with "new" numbers though, if you want to have more in the end than just a fancy binary tree - in particular if you want to map your "fake numbers" to the reals, infinity, etc.
- danabramov 14d agoThis procedure is enough. If you define addition and other operations in a certain way (as Conway did), it turns out that on the omega-th day (i.e. after initial infinite steps), all reals will be born.
- skeledrew 14d ago> position does matter Only as a mental abstraction that's based on our experience/concept of space+time.
- howunfortunate 14d agoThis is very helpful! Basically what I take away is that we're inventing a new number system from scratch. So we're not "proving" that 1 is a number between 0 and the empty set. We're defining it as such, and it just so happens that a number system defined this way works out in convergent ways with other mathematics. Is that roughly right?
- danabramov 14d agoExactly.
- deleted 14d ago[deleted]