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Suppose we want to prove a predicate in the $$HyperReal$$ Numbers,by (weak)induction,for example,for all: $$x\in\mathbb H,x^2>=0$$ Would we Proceed as follows?: $$P(dx)= dx^2>0,$$ $$P(x)=x^2>=0,$$ $$P(x+dx)=x^2+dx^2+2xdx>=0,$$ and since $x^2 ,dx^2$ and $xdx$ are all positive,then for all Hyperreal numbers this predicate holds? If wrong,What is wrong with that proof ? What kind of defintions am i missing? If right , then the can I use the transfer principle to make it true in the real numbers too?

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  • $\begingroup$ Why are you using induction? The transfer principle just applies: $(\forall x)(x^2 \geq 0)$ is true in the reals, and it's a first-order statement, so it's true in the hyperreals. $\endgroup$ – Patrick Stevens Nov 19 '16 at 19:03
  • $\begingroup$ I want to develop ("Generalize") mathematical induction,so ,if needed,I can work out something by induction and don't worry about the Constructive proof. $\endgroup$ – Logan Luther Nov 19 '16 at 19:10
  • $\begingroup$ I'm about 70% sure of the following comment: your proof is wrong, because if it were right then it would transfer directly to the reals, and it's wrong in the reals. Try working through the following (over the hyperreals) to see why: math.stackexchange.com/questions/1567075/… $\endgroup$ – Patrick Stevens Nov 19 '16 at 19:13
  • $\begingroup$ @PatrickStevens I think I see why the proof in the following link you proposed is wrong,but i think acknowledging that that stating a first right statement and then going on to proving it ,doesen't always work,for example: for all $n\in \mathbb N, n!>2^n ,$ and it only works when $n>3$. So if we fixed that part of my Reasoning,what other things would be wrong? $\endgroup$ – Logan Luther Nov 19 '16 at 19:25

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