True, You can only induce natural numbers from this.
However, you could extend it to the positive reals by saying [0,1) is a small number. And building induction on all of those.
You could cover negative and even complex numbers if “small” is a reference to magnitude of a vector, but that is a slippery slope…
In a very not rigorous way, you can cover combinations of ordinal numbers and even non-numbers if you treat them as orthogonal “unit vectors” and the composite “number” as a vector in an infinite vector space which again allows you to specify smallness as a reference to magnitude like we did for the complex numbers.
If you multiply two not really numbers, just count the product as a new dimension for the vector. Same with exponentiation. Same with non math shit like a cow or the color orange. Count all unique things as a unique dimension to a vector then by our little vector magnitude hack, everything is a small number, even things that aren’t numbers. QED.
This proof is a joke, broken in many ways, but the most interesting is the question of if you can actually have a vector with an uncountably infinite (or higher ordinals) of dimensions and what the hell that even means.
I don’t think you can induce all numbers from n+1
True, You can only induce natural numbers from this.
However, you could extend it to the positive reals by saying [0,1) is a small number. And building induction on all of those.
You could cover negative and even complex numbers if “small” is a reference to magnitude of a vector, but that is a slippery slope…
In a very not rigorous way, you can cover combinations of ordinal numbers and even non-numbers if you treat them as orthogonal “unit vectors” and the composite “number” as a vector in an infinite vector space which again allows you to specify smallness as a reference to magnitude like we did for the complex numbers.
If you multiply two not really numbers, just count the product as a new dimension for the vector. Same with exponentiation. Same with non math shit like a cow or the color orange. Count all unique things as a unique dimension to a vector then by our little vector magnitude hack, everything is a small number, even things that aren’t numbers. QED.
This proof is a joke, broken in many ways, but the most interesting is the question of if you can actually have a vector with an uncountably infinite (or higher ordinals) of dimensions and what the hell that even means.
Sure you can. Proof:
0 is a number.
If n is a number, n+1 is also a number.
Therefore, by mathematical induction, we can induce all numbers.
☝🏽🤓 The naturals are hardly all numbers, considering they’re only a countably infinite subset of the reals.
Weird. There are no numbers that can’t be added. I wanted that.