0.99999~ = 1 ?

If I've erred (possible), then please rephrase it in the form of a proof which can point out my errors; i.e., show which aspects of the following are inaccurate summations of SG's argument, and please correct them:

P "All real numbers have properties X"
P ".9999| does not have properties X"
C ".9999| is = 1 and it is false that .9999| ≠ 1"So far, you folk have just been calling me foolish and telling me my logic is faulty; no one has yet to show me why or where.

So, does the 'real' in 'real numbers' have any relation to the 'real' in 'reality'?I think the confusion has come from me assuming that it in some way does, but the more folk describe this concept mathematically, it appears that I was on the right track with my first post: Which seems in line with what (I think) Catholic Scientist and RAZD have mentioned earlier; namely, that the equality of 0.9999| and 1 is more of a result of the way in which you have defined your system; defining it differently gives us different 'equalities' of infinites to 1. In other words, you have not given a definition (as far as I can tell) of 'real number' that defines it concretely in terms of 'reality', but merely the one you've given defines it in terms of the 'established mathematical system'. (RAZD (Message 12) gave the lovely example of the equivalence of .8888| and 1 in a nonary system.)Let us plug what you've introduced back into the proofsas I understand it(here, property Z is the properties you have listed that define 0.9999|too many to list separately, we variablize them):P "All #s with property Z are REAL within MATHSYSTEM (i.e., ((R/Z)/M) (M=MATHSYSTEM)"
P "0.9999| has property Z"
P "(The REAL of 0.9999| is True) given MATHSYSTEM is True (i.e., ((R/0.9999|)/M))"
P "[proof of M=True]"
C "(R (the REAL of 0.9999|) is True... period"Proof that .9999| is not DISTINCT (from 1):
P "It is false that 0.9999| is both REAL and DISTINCT (from 1) (~(RD))"
P "(R (the REAL of 0.9999|) is True... period"
C "0.9999| is not DISTINCT (from 1)"In essence, there has been little done in regards the explanations given for why 0.9999| is REAL other than to introduce the variable, in form of caveat, MATHSYSTEM, the veracity of which (i.e., its conformity to reality, last premise in first proof) must now be shown to support the proof for 0.9999| being REAL and making the conclusion that it is not DISTINCT (from 1) unavoidable (within the world of reality, in which we all strive to live; if it is the case that the only place where such a proof has relevance is within the world of number manipulation (MATHSYSTEM), and the MATHSYSTEM is not necessarily related to the real world, then I've little interest in continuing to understand the equality of 0.9999| and 1, since it would appear little more than a novelty of an imperfect system which allows functions and manipulations that do not have reality as their basis). Of course, it may be that our conditions for DISTINCT (from 1) are also only definable given MATHSYSTEM (D/M)i.e., that all parts of the proof's conformity to reality hinge on the veracity of the MATHSYSTEM, the disproval of which bringing destruction to the proof, but for now it will be enough to focus on just the REAL of 0.9999|.So, I look forward to your proof of M=true, or the admittance that the MATHSYSTEM is not necessarily related to reality.²Jon

Excellent. So you have chosen to assert that there is a disconnect from MATHSYSTEM and reality, and without such a necessary link, you are free to claim MATHSYSTEM = True.

This is exactly what one would expect in a system which represents two identical real-world values (values from REALWORLDSYSTEM) with two different symbologies.

(That is, one would expect there to be a necessary disconnectand not a necessary relationbetween any two systems if the same thing in one system had altering representations in the other, cf. Language.)

In fact, it is this disconnect that is primarily at the heart of the necessity of the DISTINCTness criteria, which says that "for any numbers A and B with the following properties X, they are not formally DISTINCT in the REALWORLDSYSTEM, though being formally distinct in the MATHSYSTEM". This is necessary, because as you point out, without it, we could get ourselves into a paradoxical mess in which the MATHSYSTEM would start out representing the REALWORLDSYSTEM and then end up not doing so (which is fine if we do not wish the system to work, but we do wish it to, so it is far from fine).1 = 3/3 = 1/3 + 2/3 = 0.3333| + 0.6666| = 0.9999| ≠ 1So, the MATHSYSTEM introduces a function that equates 0.9999| with 1 and thereby closes the paradoxical loophole, such that we may continue using the MATHSYSTEM as a representation of the REALWORLDSYSTEM. Of course, just because we can mend one system so that it will represent another, does not indicate a necessary representative property of the former system in regards the latter, i.e., it does not show that the former system represents by necessity the latter system, but merely shows that it represents it, not necessarily by necessity.Afterall, any system that represented necessarily another system would not need a function to close paradoxical loopholes, as such loopholes would not exist. The fact that the MATHSYSTEM has introduced such a function in an attempt to represent the REALWORLDSYSTEM, shows that the MATHSYSTEM is not necessarily representative of the REALWORLDSYSTEM.But all these things arise, of course, because MATHSYSTEM is not in a necessary relationship with REALWORLDSYSTEM. Afterall, in REALWORLDSYSTEM 0.9999| v. 1 is a formality which is meaningless... how d'you say... gibberish.Jon

LOL. Whatever. When you have more to say other than "I don't get it", I will be ready to continue with you.

Hi Dr ACuriously you have just proven that you have not read my post in context.

I don't understand this.You went logically along until the very end. I don't (logically) see any reason why you put a "≠" in instead of a "=" at the end. What would make you do so?

The proof was intended to show that 1 ≡ 0.999~ by assuming that it wasn't, and then showing that this results in a contradiction.

In the process it uses another version of 0.999~ and the problem is that if one is not 1 then the other isn't either and it remains half way between. One can't use the conclusion as part of the proof eh?

What is it about math that makes people go mad, anyway? Or is it simply mad people that practice math?

Not true, there are studies that have shown a correlation between creativity and Bipolar disorders.

Now for a mea culpa. As erudite and definitive as the above sounds, there is a very good chance that I don't know what the hell I'm talking about.

I think there is a fine difference between growing to infinity and infinitely growing. I would say that the latter is possible in respects 0.9, etc., while still maintaining it has a value less than 1.0. Afterall, the number we are working with is infinite, is it not?

I see no reason why Math could not permit such infinite divisions in contradiction to Reality, which clearly does not. (Damn, there's that 'Zeno' whisper sound again.) Now, I realize that it doesn't work this way, and so long as the MATHSYSTEM maintains the DISTINCTness criterion it too will not permit such infiniteness. But that it needs a check-sub-system to eliminate contradictions with Reality should be a good sign that the MATHSYSTEM and its parts do not follow necessarily from Reality (REALWORLDSYSTEM), just that we have found a way to make the system both internally consistent and eliminate its contradictions to Reality. Anyway, more on this relationship when my dern thread gets (if ever) promoted.

It's not?

How many decimal places does 0.9999| go out to?

Infinity - infinity is undefined.