> =?UTF-8?Q?Martin_Semer=C3=A1d?= wrote: > > > =?UTF-8?Q?Martin_Semer=C3=A1d?= wrote: > > > > > > > > For example - if you have number 1,0 and other > number 0,999... in > > > > decadic system (that is 9/10 + 9/100 + ...), > then these two numbers > > > > are stricly diferent > > > > > > Proof? > > > > Ok, you can not proof it in a nowadays standard set > theory or theory > > of classical real numbers - of course. > > So, it isn't true. So, you shouldn't say it. If you > want to discuss the > non-standard reals, you should say so.
"So, it isn't true..."
It is the question. As I said before, in classical "rational theory" with operators +,-,*,/ you can not proof that there are any non-rational "numbers" on the geometric line - or one can say - you can not proof there are empty spaces or one can say "non-standard rational numbers" or irrational numbers - inside this full rational line.
So rational numbers are, of course, dense and without any voids - if we are talking only about rationals - or when we are talking about pure theory of finiteness. But - its TRUE there are voids on this line and we should talk about them and call them irrational numbers. But in the same time and in the same way we can proof also there are still voids in this real line, inside these real numbers as well. Of course it is also TRUE - and I could not only say it, but also proof it, but of course I can not proof it only with, for example, three or four digits!, I can proof it only when using true infinity. It is the same as you can not proof irrationality of the number PI even you can not talk about it, when you know nothing about infinity at all.
Bernard Bolzano proof that 0,999... is different number from 1,0 120 years before there were any non-standard set theory. Also Leibniz knew about such numbers. And of course Zenon did.
My conclusion is following: at first you have to understand, why there is NO empty space in the classical reals. That is a wonderfull story! But in the next step you have to go further and discover there are still empty places on this line - which are fill up with non-standard numbers or one can say also with infinite small numbers.
As I said before - wirst step is to understand, why sqrt(2) is NOT an rational number. In the second step, on this higher level, you can learn how to construct such natural numbers P and Q that satisfied: P/Q = sqrt(2).
At the first level you have to understand why reals are bigger than naturals in the way of Cantors cardinality. It is a wonderfull story. And after it you can go and understand the proof of the opposite.
Firstly you have to understand how to walk on the Earth, secondary how to fly in the Air. May be you know about French academy - when they postulate as true there are no rocks on the heaven. Of course this is true - for a long time, till there is no falling star on the sky.
I am not sure, if you are satisfied with my unswer. Of course, if you are over fifty, you can have problem with this. My children have not, yet...
