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If we look up the definition of induction, that is by showing the base case and statement n => n +1 then we would get ever single natural number proofed for what ever we desire, more over , this should work for every system that is close under successors. So my question is that can we somehow defined a very base value that is more than limitlessly small and use the property of induction to proof every positive real number? (even given that there is no smallest real number, can we do things like $ n => n +r$ for $r,n$ belongs to real number? (r does not need to be the smallest number this case, i suppose that there should be a way we can claim r can be every real number we desire) and from Even more over, can we also defined negative number as a system close under successor so that induction would work for all real numbers? Thank you for answering the question!

Hanul Jeon
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    "can we somehow defined a very base value that is more than limitlessly small and use the property of induction to proof every positive real number" $;-;$ No, see for example Prove that there is no smallest positive real number, Why there is not the next real number?. – dxiv Jan 03 '22 at 07:38
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    There is however, something called "real induction", and it is quite pretty https://arxiv.org/abs/1208.0973 – Calvin Khor Jan 03 '22 at 07:45
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    After the edit: $;$ "can we do things like n=>n+r for r,n belongs to real number?" $;-;$ You sure can, and they don't even need to be real numbers, you can restate it in very general terms over arbitrary sets. Point, however, remains that induction can only prove a statement over a countable set. But real numbers are not countable, not as a whole and not even the tiniest real interval $[a, a+\varepsilon]$ is countable. – dxiv Jan 03 '22 at 08:02
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    @dxiv To be a little more precise, you need a well-ordering on your set to use induction on it, as long as you can use transfinite induction. But if you have that tool in your toolbox, you can use induction on arbitrarily large sets. – Robert Shore Jan 03 '22 at 08:15
  • @RobertShore Right, though transfinite induction feels a bit beyond the scope of the question here. – dxiv Jan 03 '22 at 08:36

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So my question is that can we somehow defined a very base value that is more than limitlessly small

No, because "a value that is more than limitlessly small" does not exist in the real numbers.

5xum
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