Answer: yes.
It’s just proving it that’s a bitch!
I was curt for being talked down to with a Zeno for Dummies link, but I accept the criticism for being impatient.
I certainly don’t want to revive an old, 23-page thread, but has anyone presented the actual definition of the real numbers (say, in terms of Dedekind cuts or as the completion of the rationals, and as opposed to constructions like infinitesimals)? I’ve only read about 2/3 of the thread and am…skeptical…about the ability for a fairly involved argument to convince anyone who vehemently denies that .999… = 1; but I’d be happy to plow through it if anyone cares.
See post 36, in which the notion of Dedekind cuts of the relevant sort is implicit, yet presented very simply.
Sure, that works.
[quote=“Cognitive_Tide, post:1111, topic:27517”]
9 x (1/9) = 1 = 9 x 1.222… + 9 x [.111…]^2
[/QUOTE] The video is right, the equation in your title is not. (9 * (1 + 2/9)) + (9 * (1/9)^2) = 11 + 1/9.1 ≠ 9 x 1.222… + 9 x [.111…]^2 => (9 * (1 + 2/9)) + (9 * (1/9)^2) ≠ 11 + 1/9
Proof for .999… ≠ 1
Assume .999… = 1,
Use this assumption via substitution to arrive at a clearly
erroneous result by applying the axioms of the system within the system
3 x .111… = .333…
=> 1/3 = .111…/.333…
1/3 = 10 x .111…
-------------
10 x .333…
1/3 = 1.111…
-------------
3.333…
1/3 = 1 + .111…
-------------
3.333…
The key step, square both sides to bring the infinitesimal out of hiding
1/9 = [1 + .111…]^2
---------------
[3.333…]^2
The most critical step in the whole proof. Stopping and considering what the square of something clearly less than one COULD be…it was never clear that the square of .999… could not be one - but it IT IS clear that the square of .333… cannot be 1(in other words it is clearly not the same as the whole but rather a part of the whole)…if you are not convinced of this simply think of squaring the fraction 1/3 to get 1/9, all that you need to notice about the squaring of 1/3 is that it results in the much smaller magnitude of 1/9, it is impossible to demonstrate this relationship with .999… and 1 because the radix of the system is 10. it is this conceptual step that allows us to square the mantissa and separate the characteristic from it for the remainder of the proof. Also notice the notation indicating the squaring of 3 along with the square of [.333…], whatever the square of .333… is it would be added to 9, however it will be eliminated before the answer to that is required
1/9 = [1 + .111…][1 + .111…]
--------------------------
9[.333…]^2
1/9 = 1 + .111… + .111… + [.111…]^2
-----------------------------------
9[.333…]^2
1/9 = 1.222… + [.111…]^2
----------------------
9[.333…]^2
9x(1/9) = 1 = 1.222… + [.111…]^2
--------------------------
[.333…][.333…]
very important step of multiplying through by 1/3 because it will allow for subbing in 1 for .999…(twice) in the denominator
1/3 x 1 = 1.222… + [.111…]^2
----------------------
3 x [.333…][.333…]
1/3 = 1.222… + [.111…]^2
----------------------
[.999…][.333…]
1/3 = 1.222… + [.111…]^2
-----------------------
[.333…]
Multiply by 1/3 again to eliminate denominator
1/9 = 1.222… + [.111…]^2
Multiply by your favorite integer 9
9 x (1/9) = 1 = 9 x 1.222… + 9 x [.111…]^2
I have more like these.
Seen enough?
Since when does
[3.333…]^2 = 9[.333…]^2 ???
Every step you took is correct except in turning “[3.333…]^2” into “9[.333…]^2” (these being the denominators before and after your passage about “The most critical step”).
It seems your reasoning for this is something like the assumption that [A plus .BCDEF…]^2 = A^2 times [.BCDEF…]^2, but that’s not actually true.
What IS true is that [A times .BCDEF…]^2 = A^2 times [.BCDEF…]^2, but that’s different than the move you are making.
What may be throwing you is that the "plus"es and the "times"es above are both often notated by simple juxtaposition (that is, without writing explicitly “plus” or “+” or whatever, or “times” or “x” or whatever, instead simply writing the two operands next to each other), but they are in fact different things.
(Actually, it would be rare to write “A.BCDEF…” to mean “A times .BCDEF…”, precisely because of the conflict with the much more common reading as “A plus .BCDEF…”)
Actually, this way of putting it may be clearer:
Every step you took is correct except in turning “[3.333…]^2” into “9[.333…]^2” (these being the denominators before and after your passage about “The most critical step”).
It seems your reasoning for this is something like “[3.333…]^2 = [3 x .333…]^2 = 3^2 x [.333…]^2 = 9[.333…]^2”.
But that first part, where “3.333…” is taken to mean “3 x .333…”, doesn’t actually hold up; yes, we often denote multiplication simply by writing the two factors next to each other, without an explicit “x” or “times” or what have you in the middle, but that’s not what was meant by writing “3” next to “.333…” in this case (as seen from how you originally got “3.333…” as “10 x .333…”).
Three thirds (that is, 3 * one-third) = 1.
Thanks for pointing that out so quickly and clearly Indistinguishable, looks like I saw conceptually what I wanted to see there and made a conceptual error.
One more question though:
The incompleteness theorem (if I understand it correctly) states that it is impossible to create a system which describes itself with a finite number of statements unless the system contains at least one statement that is not provable within the system. The statement .999… = 1 appears to me to be that statement. Applying the axiom of subtraction the statement is equivalent to 1 - .999… = 0, which is not provable (again the system MUST have at least one axiom that is not provable to satisfy the incompleteness theorem so 1 - .999… = 0 therefore should not be provable. However the statement 1 - .999… = <unknown> is also not provable. Wouldn’t the latter statement be a better form of the axiom than 1 - .999… = 0
I have heard the one about the nonexistence of nonzero infinitesimals but how do you integrate using the real number system and zero for the differential?
BTW, if you or anyone here has any thoughts on why it is not possible to use .999… in the Lorentz equation to arrive at a meaningful result I would like to hear them.
1 / 3 = 0.333…
3 * 1 / 3 = 3 * 0.333 …
1 = 0.999 …
How is that not proof?
The problem with using that as your “statement” seems to be that 1 = 0.9999~ has been proven in a thousand different ways and is accepted as proof by every mathematician.
Other than that, of course…
You have the incompleteness theorem sort of backwards. It isn’t that any system must have an unprovable proposition in order to be self describing - it is that any such system will have unprovable propositions. The implication runs the other way. You don’t have to find the unprovable propositions, they come for free. Moreover, they tend to be evil issues that revolve around the nature of the self describing mechanism itself. They aren’t just some mundane bit of arithmetic.
A minor, but important nitpick on terminology as well. All axioms are unprovable - they are the things you must assume are true as the basic building blocks from which the rest is built. Conventionally arithmetic starts with Peano’s five axioms defining the integers. You get a long way with just them. Operations on the integers are not axioms, they are definitions, and using them you can quickly build up most of number theory.
Similarly you can build up a useful set theory from a few axioms. It gets more fun when you get to the point of worrying about adding the axiom of choice.
The two implications you are describing run the same way… “Any X must have a Y” and “Any X will have a Y” (along with Cognitive Tide’s “It is impossible to create an X unless it has a Y”) are basically trivial rewordings of the same mathematical claim.
But it’s right to note, as you do, that the Incompleteness Theorem doesn’t just propose the existence of some random unprovable proposition which one can then go hunting for. In fact, it is quite constructive: it tells us a very specific proposition which will be unprovable (or, rather, a property such that no theory can consistently prove it has that property). It has no implications for the provability or lack thereof of any other, generic propositions, except insofar as they might relate to that one.
Right on. That having been said, there are people who would disagree. It is very common to see even sophisticated logicians claim such things as the independence of the continuum hypothesis from ZFC to be manifestations of the GIT phenomenon. But they are wrong; GIT has nothing to do with it. They are just other examples of some theory not entailing some proposition, a not particularly rare phenomenon.
It seems to me Cognitive Tide, in using the language of “provable within the system”*, was speaking perfectly cromulently. It is standard to use “provable” in such a way as to count an axiomatic theory as proving its axioms (“A is provable (from theory T)” meaning simply that there are axioms within T which entail A).
[*: That having been said, Cognitive Theory also speaks later of “axiom that is not provable”, so I’m not certain exactly how he is using the language of “provable” and “axiom”. It doesn’t really matter.]
Glad to have been of help.
The incompleteness theorem is irrelevant here, as noted above. Regardless:
You have made some claims about “.999… = 1” and equivalently “1 - .999… = 0” being unprovable within “the system”, without ever precisely specifying what “the system” is, and why we can be confident it does not prove these. I would note that there very much are a number of mathematical frameworks which DO prove these propositions, should one want to do so. Most trivially, we could simply consider a framework in which one of our basic definitions is that “.999…” is to be interpreted as equal to “1”, by fiat; perhaps you would find such a system uninteresting or poorly motivated. In which case, you can consider any of the other arguments given in this thread to motivate the identification, and note that they can be formalized in various frameworks as well.
That having been said, there will also be frameworks which fail to prove this, or even prove some other thing like “1 - .999… > 0”. As noted previously in the thread, we have, here as everywhere else in mathematics, a choice as to how to interpret our notation, and could choose, if we liked, to use the notation of “.999…” to mean various different things. Until we actually specify a particular formal system, there’s no point saying that it does or does not prove a particular claim about “.999…”, and even then, we can always consider other formal systems just as well.
[But standardly, mathematicians do use “.999…” to mean “1”. This is a definition they have settled upon which is convenient to them. See post 36.]
Are you asking “Usually, when one integrates, there is an idea of summing up infinitely many infinitesimal quantities (e.g., note the ‘dx’ in ‘the integral of x^2 dx’); how does one make sense of this in a framework without nonzero infinitesimals?”?
It is possible to rephrase all language of “infinitesimals” in such a development of calculus into language of “for small (but not infinitesimal!) such-and-such, we get an approximate result which is only off from the ideal by so-and-so, where the so-and-so can be made as small as we like (short of zero) by making the such-and-such sufficiently small”; if this sort of translation away of infinitesimals (sometimes called “(delta-)epsilontics”) appeals to you, you will find it developed in any standard modern book on analysis (and given lip service even in any modern textbook on introductory calculus).
That having been said, if, like the originators of the subject, you like using infinitesimals to understand integration, there’s nothing wrong with that (even if your teacher says there is!), and in some sense they’re lurking underneath anyway even if one does not formally reify them. Just because one may sometimes choose to restrict attention to one sort of quantity doesn’t mean one cannot use other sorts of quantities for other purposes. Go ahead and use infinitesimals to your heart’s content.
But it has nothing intrinsically to do with what “.999…” means, which is a question of NOTATION.