> You seem to think that the real numbers derive their justification from > geometry ( the real number line?). There are at least two totally > non-geometric justifications, via Cauchy sequences of rationals or via > Dedekind cuts of the rationals, neither of which depends on geometry, > and in both of which the proof that 0.999... and 1.000... must be > interpreted as the same real number. >
Cauchy sequences and Dedekind cuts are not something about which I can speak intelligently. Thank you for pointing them out as other ideas to understand real numbers through.
> Then you are not talking about the real real numbers, which your > interpretation of 1 - 0.999... is prohibited by the Archimedean property > of the reals.
This wouldn't surprise me as I don't consider 1 - 0.999... to be a real number.
> > > No other real number can keep from exceeding one > > a finite number of its increment, let alone an infinity of them. > > > You have said a > (1-(1/10^n)) for any natural number n and this is > > indeed the case. I agree that 1-(1/10^n) is a real number for any real > > number n. Therefore 1 - a will be a real number. 1 - a will also > > always be less than 0.999... for any paricular real number n. > > Therefore, the needed subtraction to obtain 0.999... from one, should > > a relation exist, may not itself be a real number. > > So suddenly, the reals are not closed under subtraction? > That is another property impossible in the standard real number system.
I have said I agree that the reals are closed under subtraction. I have said that that which we would have to subtract from 1 to yeild 0.999... would not be real.
> > > > But it is trivially imperatively that 0.999... is real in the Cauchy > sequence model, and while somewhat less trivial, it is also imperative > in the Dedekind model.
I cannot account for the needs of other systems. I would be very interested and curious to explore the relation of all these ideas, and I thank you for you continually expansive introduction of other ideas as I feel they all add.
Yet as we are aware, mathematical systems can always be created that allow something of our chosing as we have the luxury of definition.
I claim only that real numbers themselves do not allow 10*0.999... - 0.999... to be equal to 9, but strictly somehow less, whether we can express that number in our current numbering systems or whether that number itself is static and real.
> > > What it demonstrates is that if x = 0.999... then 10x - x must > > be less than 9 within the confines of multiplication upon real > > numbers. > > The difference, whatever "size" it may be, must be a real number, as > both terms of the difference are, and according to you, must > simultaneously not be a real numbers.
I do not place the same faith you do in the idea that 0.999... and its distance from one are real. I call them dubious numbers and I have not fully explored their true relation to the real numbers - this is exploration in progress in my own studies. I am simply unable to call the distance real because it would take an infinity of them to equal one. This is my prime motivation for calling the distance "non-real". I have notice that it takes two types of number to add up to a whole number. To get a whole number from an irrational you seem to need an irrational. From this observation I note that through my current understandings this distance between 0.999... and 1 is not real, so I find that 0.999... should be equally unreal.
But note, the distance may in fact be entirely real - In the future I will share what I come to understand. Anything between now and then will be the rambling mind of someone rifling through a infinite set of possibilities.
> > > What the proof demonstrates is that subtraction is required > > and omitted. > > it is clear by now that your proof does not hold for any standard model > of real numbers, so you must be talking about some private number > system of your own that no one else uses.
Indeed I am not. Real numbers cannot allow a result of 9 after an operation of 10*0.999... - 0.999... - they need it to be less. Not because they have a real value they can themselves incorporate to yield the answer, but because it infinite operation applied to the set of their partitions times 10 minus their paritions is never subtracting an appropriate ammount from the final result. It must be less.
> > > It would not be surprising that the needed subtraction is > > not real. > > Not to me. > > > Real numbers simply will not allow 10x - x to be 9, > > If x = 0.999... = Sum(9/10^n: n in N), the real numbers I use will not > allow it to be anything else.
Awesome. I don't understand why you would need to assume its real. I currently cannot take it to be so. I can only take it to exist within relation to all other real numbers as strictly equal, less, or greater.
> > they > > > need it to be somehow less, whether the less-ness is describable in > > real numbers or not. > > The difference between any two reals must be a real, at least if one is > to have the standard field of reals.
I use this same definition. I also note that in using the definition, I would have to allow it to classify my other number.
My prime reason for even suggesting that 1 - 0.999... would not be real is because it would take an infinity of them to equal 1, and a real number could not accomplish this.
> > Well, you have certainly presented us with one that is quite dubious > enough to go on with.
I agree. I only wish to state one more time that I do not hold that real numbers can allow 0.999... to equal one. If you can account for why we would need not incorporte the missing subtraction that I point out, you may change my whole perception of this topic.