Is this a valid method to construct N if you're given R?

Is this a valid method to construct N if you're given R?

{x ∈ R : x > 0, floor(x) = x}

floor(x) can't be defined without using natural numbers or integers. It's better to write {x ∈ R: x>0, sin(πx)=0}

How would you define sin?

y''+y=0

Isn't that recursive definition?

why would it be?? you are already giver R.
It is incomplete tho, all harmonics solve that equation, you need something like y(0)=0 y'(0)=1

how would you define \pi then?

inf {x ∈ R | x > 0, sin(x) = 0}

just use power series brainlets

Oh ok that seems to work.

Power series is probably the simplest way.

If you are given R you have 1 since it is a field. Constructing N from 1 is easy..

are you retarded?

Power series requires factorial and so requires N

R is defined using N, what are you guys smoking? Can I have some?

Yes, but how would you define N if you had R and R somehow wasn't defined using N?

what the fuck you brainlets

take the elements 0 and 1 and 1+1 and 1+1+1...
how is this hard?

i cant into set builder but it would look something like this

x goes into R : x_0 = 1, x_n = x_(n-1) + 1

Let x be any non-zero element in R.
Then we define 1 to be x/x.
The define any other natural number to be 1 plus some prior one.

Let the multiplicative neutral element be a natural number.
Let the sum of two natural numbers also be a natural number.

It should be noted however, that even though we may have some prior definition or understanding of the Real numbers, this construction lies on the idea of sums and a multiplivative neutral element, which we haven't defined.
It goes the same with OP's construction, which uses some function "floor" which is undefined.

Smallest additive monoid containing 1 lying in R

You only need a succesor function to prove peano's axioms with this , but i have no idea of how to define such function in a formal fashion.

I meant this 1+1+1 ..