0.99999~ = 1 ?

Your logic goes like this: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"Somewhere the logical connection is missing, as your conclusion does not follow from your premises; perhaps you can explain it to me better if I misunderstood. So far, it looks like you've just proven .9999| not to be a real number, which seems more a matter of definition than a matter of real-world fact.So, I am open to believe you; just provide supporting evidence of ".9999| is = 1 and it is false that .9999| ≠ 1"I await your modification to your proof Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

LOL. Again, you base your conclusion on the premisethough, I give you extra credit for more cleverly disguising it this timethat .9999| is a real number. Your flow can just as easily be reworded as follows (separating the adjectives out):P "All numbers DISTINCT (from 1) have property X₁
P "All REAL numbers have property X₂
P "All numbers that are both REAL and DISTINCT have property X (i.e., X₁ + X₂)¹
P It is false that .9999| has property X (i.e., that it is both REAL and DISTINCT (from 1))
C It is false that .9999| is not both REAL and DISTINCT (from 1)You have proven that REAL & DISTINCT (from 1) do not together describe .9999|, but this only proves that .9999| is either not REAL or either not DISTINCT (from 1) at minimum. Occam's razor tells me to only accept the minimum, which means I need only accept one or the other of your (two) conclusions. The problem is that you've given no reason to accept the REALness of .9999| in place of its DISTINCTness (from 1). So, your reasoning, as it stands, still allows the possibility of .9999| being DISTINCT (from 1)i.e., it does not prove .9999| to not be DISTINCT (from 1), as all the reasoning given thusfar is still valid if .9999| is DISTINCT (from 1) and not REAL, in other words, it satisfies the conclusion that .9999| not have property X. For your reasoning (and mine) to be accepted, only one of the properties has to be false for .9999|, i.e., it either has to be not REAL or has to be not DISTINCT (from 1). So, you need a separate proof that tells us .9999| is REAL, leaving us to accept only DISTINCTness (from 1) as the property that must be false to satisfy your (and my) original (and accepted-by-both-of-us) reasoning.Thus, my initial challenge, which was for someone to prove .9999| to be a REAL number which, as the logic shows, could NOT be true if DISTINCTness (from 1) were also true. The full proof(s) will look like this:Proof that .9999| is REAL:
P [P1 for .9999| being a REAL number]
P [P2 for .9999| being a REAL number]
etc.
C [Conclusion that .9999| is a REAL number]Proof that .9999| is not DISTINCT (from 1):
P1 "It is true that 0.9999| is REAL"
P2 "It is true that 0.9999| is DISTINCT (from 1)"
P3 "It is false that both P1 and P2 are true (~(P1P2))"
P4 "[conclusion from proof of 0.9999| being a REAL number], i.e., P1 is true"
C "P2 is false, i.e., .9999| is not DISTINCT (from 1)"Note that premises 1, 2, and 3 are just an expanded restatement from the first proof (beginning of post), with premise 4 (0.9999|'s REALness) being the one we need to insert to make the conclusion that .9999| is not DISTINCT (from 1) unavoidable.In short, following your (and my) reasoning, there is no way to prove .9999| to be equal to 1 without first showing it to be a real number. Of course, I have no idea if .9999| is a real number or not, and look forward to anyone who will prove that it is and thus prove .9999| = 1. (I just pray to God that the definition that gets used for 'real' number be not so pitiful that anything qualifies.)Jon{ABE: I noticed you edited your post after I started typing my reply; worry not, though; italicizing your conclusion doesn't affect its truth value}
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¹ X₁ and X₂ are, of course, not necessarily separable properties, but are used to show the necessarily inseparable relationship of the first two premises to the third (i.e., property X only appears when properties X₁ and X₂ have both been satisfied). All three premises could be rewritten as: "When REAL and DISTINCT (from 1) are both True, then X is false", which would require the addition of "X is True", and the conclusion "REAL and DISTINCT (from 1) are not both True (when said of .9999|)". The problems with this that follow would all still be the same.Edited by Jon, : LOL

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

I guess my prayers were not answered. Can you provide a definition of 'real number' so that I do not have to look for one myself?

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

LOL. I'm too busy for malarkey. My debate is with lyx2no, now. If you want to join, you can start by supporting this latest claim.Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

LOL. Yes; too busy for malarkey, but I can make time for bullshit. If you wish to be obtuse, care to point out where in my rewording the meaning of your original was lost?Edited by Jon, : Mr. T

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

Rrhain,Thank you for actually participating with reason and evidence. My math is a little rusty; can you explain this one to me? What does it mean for numbers to converge upon e?Thanks,
Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

I did not think 'rational' and 'real' numbers to be the same. One's a subset of the other, no? Which are...?

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

How is this different from saying:Real numbers are numbers that meet definition X
Real numbers are defined by X
All real numbers are RealWhat purpose does the Real/Non-real distinction serve? Does it actually differentiate, or is the distinction merely made up?

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
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¹ It should probably be mentioned now that using Unicode (UTF-8) would result in the best display with all these symbols.
² If this is going too far off-topic, we can start another thread; just give the word, Admins.

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

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. Is this to say that the argument he has given for ((0.9999| = 1) and ~(0.9999| ≠ 1)) is only relevant in the world of mathematics?I look forward to your active participation. Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

I was under the impression that infinite numbers (which is what you get if you subtract 0.9999| from 1, no?) never have to stop. When we stop it, we are admittedly not working the the number proposed, which was infinite, and if we leave it be infinite, then we can never assess whether it is equivalent or not (can we?). In other words, there never seems to be a way to link the mathematical 0.9999| with anything in reality that would make it meaningful, such that asserting that a number like 0.9999| exists always appears to be a useless triviality.So, is there any reason to assert the existence of 0.9999| other than to dazzle the Kindergartners?

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This has been my suspicion from early on, as I said in Message 87: I wish folk would have just agreed with me then that 0.9999| had no real-world significance and was instead just a fancy mathematical parlor trick.Interestingly, if the other folk would follow my proofs through (instead of dismissing them without reason), particularly the ones in Message 110, they would see that for 0.9999| to be a REAL number, and for 0.9999| to not be DISTINCT (from 1), that MATHSYSTEM would have to be such that it is not necessarily related to reality, which is a conclusion of my proofs summed up in your statement: In short, one must accept that either: 0.9999| is not a REAL number; 0.9999| is DISTINCT (from 1); or that the MATHSYSTEM has no necessary relationship to reality. In length, if one believes the MATHSYSTEM to have a relationship to reality, then looking for that link will prove M=False, which allows 0.9999| to not be REAL, which fails to prove 0.9999| = 1. However, once we accept that MATHSYSTEM has no necessary relationship to reality, then we can make M=True without second thought and 0.9999| will have to be REAL, which means that 0.9999| being DISTINCT (from 1) will have to be false, upholding our claim that 0.9999| and 1 are the same. In other words, our proofs or 0.9999| = 1 will not work unless we accept the MATHSYSTEM-REALITY link to be arbitrary; as much as the link between a word and its meaning, there is no necessary link between a number and the reality it attempts to describe. I expect the mathematicians to disagree. I did; thank you Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

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.JonEdited by Jon, : A is for: Indefinite article

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

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

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

Yes. I was 'quoting' (sans-any-reasonable-indication-that-it-was-a-quote ) the paradoxical hypothetical that Dr. A had come up with (Message 119). My point was as Dr. A's point; in a system where 0.9999| ≠ 1, there is a paradox.My extension of this was that Mother Nature doesn't have paradoxes in the first place that she must then fix with some 'distinctness' criteria that closes the paradox by equating the apparently unequal. But, I will not go further into this here, because I have started a (still un-promoted) topic (Message 1) which is meant to address the relationship of our Math system to Reality.The rest of your post seems to have been based on your understandable misreading of my horribly ambiguous statement, and so I think we can discount it.Thanks,
Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin