Above I challenged @Metaphysician Undercover to give rigorous proof for some of his claims. Unkind enough to put it in terms of "Put up or shut up," my ungraciousness.

And he did. I copy it here in its entirety:

I've already given my proof — Metaphysician Undercover

QED.

And this is not the first time MU has proved his claims in this manner. It is part of his standard form. I leave it to gentle reader to evaluate the quality of his rigorous proof as given. And I remind myself that there is no helping those who won't be helped.

The sorry fact is, that we cannot either describe or simply cannot understand infinity as clearly as we would want. — ssu

Sure, yet we do know some things at least, and can reason to some extent if careful.
Don't just ∞ × ∞ - ∞ - 7 + ∞ / ∞ + 3 / ∞, for one.
The amount of naturals isn't a natural, for another.
Maybe ∞ could be said to be a quantity that's not a number.

Much ado about very little. : — jgill

(y) (I'd hit "Like", but this will have to do)
Actually, that's a good lot of philosophy right there. ;)


Saw something fly by about adding zeros, but:
$\sum_{n \in \mathbb{N}} 0 \ = \ 0$

Couldn't see this merged thread. Maybe it went missing or was cancelled or something.

This is a ramification of the contents of the thread: 0.999...= 1 started by @jorndoe. Thanks jorndoe

First off, the simplest infinity that we know of is the natural numbers: N = {1, 2, 3,...}. The infinite set N is generated by the simple iteration of adding one to the preceding number as so: 1; hen 1 + 1 = 2; then 2 + 1 = 3; and so on and so forth.

What I'm particularly concerned about is the ratio between consecutive elements in the set N. The ratios look like below:

1) 1; there is no ratio here as there is no natural number that precedes 1
2) 2; the ratio is 1 : 2 = 0.5
3) 3; the ratio is 2 : 3 = 0.666...
4) 4; the ratio is 3 : 4 = 0.75
5) 5; the ratio is 4 : 5 = 0.8
6) 6; the ratio is 5 : 6 = 0.833..
7) 7; the ratio is 6 : 7 = 0.851742...
8) 8; the ratio is 7 : 8 = 0.875
9) 9; the ratio is 8 : 9 = 0.888...
10 ) 10; the ratio is 9 : 10 = 0.9
11) 11; the ratio iss 10 : 11 = 0.90...
.
.
.
3000) 3000; the ratio is 2999 : 3000 = 0.9996...

37896544) 37896543; the ratio is 37896543 : 37896544 = 0.9999999736123695078896904169075

As you can see as the numbers get larger the ratio between a natural number x and its successor x + 1, given by x : (x + 1) approaches, in the limit, 0.999...

But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1

In other words, there will come a point in the sequence of natural numbers where a natural number x and its successor will have the relationship x : (x + 1) = 0.999... but since 0.999... = 1, x : (x + 1) = 1 and that means x = x + 1 which basically means there's a natural number which will not increase in size when you add 1 to it. We can't say that x is infinite because if it is then x : ( x + 1) = infinite : infinite which is undefined and can't equal 0.999... Ergo x must be a finite natural number but since adding 1 doesn't get us a number larger by 1, it follows that there is a largest natural number.

Replies to this post are to be addressed to @TheMadFool

To simplify:

$\lim_{x \to \infty}(\frac{x}{x+1})=1$

How you choose to define "number" has no bearing on whether or not 0.999... = 1. Mathematical equations do not depend on what English (or French, or Swahili) words mean. Mathematics is its own language with its own terms and rules.

The below is a proof that 0.999... = 1.

$\begin{align}0.\overline{9}\stackrel{def}{=}\lim_{n\to\infty}\sum_{k=1}^{n}\frac{9}{10^k}=\lim_{n\to\infty}(1-\frac{1}{10^n})=1-\lim_{n\to\infty}\frac{1}{10^n}=1-0=1\end{align}$

To prove it wrong you need to show that its definitions are wrong (within the domain of mathematics, not within the domain of British English) or that its inferences are invalid.

Further examples can be found in the God almost certainly exists thread, in which @Devans99 repeats his OP rather than address the counterarguments.

Saw something fly by about adding zeros — jorndoe

I think you're referring to the discussion I was having which was in ℝ.

↪TheMadFool — Michael

"What I'm particularly concerned about is the ratio between consecutive elements in the set N"

n/(n+1) = 1/(1+1/n) -> 1/(1+0) = 1

Rest easy, mate. Time for a toddy.

I didn't claim this is a problem, that was Pitzotl''s misinterpretation. I said that if the same thing has two distinct names, there is a reason for that. — Metaphysician Undercover

Would you agree that the fact that a thing has more than one nam is no argument against the two expressions or representations designating the same thing?

I thought you were making that argument earlier but now you don't seem so sure.

In any event, I offer you this. .999... = 1 is a theorem of ZF set theory; for exactly the same reason that the knight moves the way it does in chess. There is no "truth" to the situation; rather there are only the rules of a formal game. If you made different rules you could defined .999... to be 47 and you could make the knight move differently. it would be fine.

The acceptance that .999... = 1, and of the consequences of the ZF axioms in general, is based on utility. When we accept ZF we can build up most of known mathematics and provide rigor to what the physicists do (usually a century or two after they've already done it). That's a pragmatic argument for accepting the axioms.

If you want to say that .999... = 1 offends your sensibilities, you are free to do that. As long as you are willing to grant the proposition that .999... = 1 is a theorem of ZF. That is a matter of objective fact that could be verified by a computer program. That is, there's a finite sequence of verifiable steps from the axioms to the conclusion.

That's really all it means; and even if you think that somehow "deep down" the equality is false; you must still admit that the statement follows logically from the axioms of ZF. So that you'd have to conclude either that ZF is inconsistent (which as far as we know, it may be) or that it's simply the wrong set of axioms for mathematics. In which case you're free to propose different ones.

:up:

Your tightening up of the mathematics is exemplary. The result will be to show in even greater relief that this is a thread about @Metaphysician Undercover, not about maths.

A ninth is the specific particular quantity corresponding to dividing one into nine equal units. — InPitzotl

You don't seem to understand. "One" does not represent a quantity which can be divided. Any multitude such as two, three, or four, can be divided, because being a multitude means that it is composed of parts and therefore can be divided into those parts. If one could be divided, then you are saying that it is made up of parts and is therefore a multitude, and not one. If it could be divided in two, then you are saying that it is made of two parts, but that would mean that it's two, not one. If it could be divided in three, then that would mean that it consists of three parts, and is really a quantity of three.

If you think that the quantity represented by "one" can be divided in any way that you please, then you deny the meaning of "one" as a single thing, because you are saying that it's really a multitude of as many things as you want it to be, existing as a unity. But that's nonsense, because that's what the other numbers represent, multitudes which have a quantitative value. If you say that one can be divided any way you please, then you are saying that "one" represents a multitude with no particular quantitative value, it is however many things you want it to be. But that's nonsense, because we all know that "one" represents a single thing, not a multitude of however many things you want it to be.

That's quite interesting. What I was saying here is a direct analog of your points about fractions and pie applied to money according to my best assessment of what gibberish you're trying to push. So if you yourself don't understand this, maybe you should heed the advice you're trying to give me. — InPitzotl

It appears you just haven't taken the time to understand what I was saying.

What are you talking about? A whole pie is one pie, not nine pies, eighteen pies, or twenty seven pies. You mean groups. Taking a particular quantity of equal sized groups is just multiplication. If I were at a farmer's market and they had a carton of a dozen eggs, I might could barter getting one half of a dozen. He'll give me six eggs. Or maybe I need more... maybe I need two dozens. He'll give me 24 eggs. Even your precious one dozen is twelve eggs. You're choking on multiplication. — InPitzotl

Again, it appears you haven't taken the time to understand what I was saying. As a result, I have no idea what you're talking about. No one mentioned multiplication, the issue was division.


Concise Oxford,1990, p813.

That the mathematical definition of "number" changes like the weather is good evidence of what I've been arguing. We do not have any logically rigorous definition of "number", and mathemagicians just use the term however they please, referring to whatever they want as "a number".

Meta did not directly address this, or any other such proof. Instead he went to an irrelevance, his claim that 1/9 is not a number. — Banno

The issue, as I said, is the op's question: "As a matter of representing numbers, wouldn't most be fine with...". I'm not fine with it, because as I said 1/9 is not a number,.it is a ratio. The op doesn't ask for proofs or any such nonsense, it asks if you are satisfied with that way of representing numbers.

Even if 1/9 were not considered a number, the proof would stand. — Banno

I couldn't care less about the proof. The op asks, "as a matter of representing numbers, would most be fine with...". If 1/9 is not a number, then we ought not be fine with this, as it is presented as a representation of numbers. If 1/9 is not a number then the presentation, as a representation of numbers, is false.

Which premise is false? — Michael

That 1/9 is a representation of a number.

How you choose to define "number" has no bearing on whether or not 0.999... = 1. — Michael

The question of the op is are you fine with this, as a representation of numbers. It is not, do you believe that .999...=1. I really do not care whether .999...=1 or not, or how many proofs there are concerning this. I'm concerned about the question of the op, is this an acceptable way of representing numbers.

So the issue I've pointed to is whether 1/9 is a representation of a number, or not. I've argued that it is a representation of a ratio and therefore not necessarily a number. Some ratios are impossible to represent as a number. That is where we get the term "incommensurable".



My point was that ".999..." has a different meaning from "1". InPitzotl insisted that it is two names referring to the same thing. Clearly it is not, because .999... is derived from 1/9 in the op, and 1 has a simple meaning without any such baggage.

The result will be to show in even greater relief that this is a thread about Metaphysician Undercover, not about maths. — Banno

Flattery will get you nowhere.

You don't seem to understand. — Metaphysician Undercover

Indeed; none of us understand, Meta. No one but you.

Now, what does that imply?

My point was that ".999..." has a different meaning from "1". InPitzotl insisted that it is two names referring to the same thing. Clearly it is not, because .999... is derived from 1/9 in the op, and 1 has a simple meaning without any such baggage. — Metaphysician Undercover

I could prove from first principles that .999... and 1 refer to the same real number. You choose not to engage with the argument. Nowhere to go with that.

So the issue I've pointed to is whether 1/9 is a representation of a number, or not. I've argued that it is a representation of a ratio and therefore not necessarily a number. Some ratios are impossible to represent as a number. That is where we get the term "incommensurable". — Metaphysician Undercover

But $\frac{1}{9}$ is commensurable. It's the fraction of two integers.

Also it seems to me that what you call "numbers" mathematicians call "natural numbers" (or maybe "integers"; do you consider negative numbers as numbers?). There's more than just natural numbers in mathematics; there's rational numbers that include the commensurable fractions like $\frac{1}{9}$, real numbers that include irrational numbers like $\sqrt{2}$, and more.

I don't see what purpose there is in saying that non-natural numbers aren't numbers, and latching onto the OP saying "as a matter of representing numbers" completely misses the point of this discussion. @jorndoe wants to show that 0.999... = 1. That's what the attached PDF tries to show. Whether or not you want to call 0.999... a number is irrelevant.

:grin:

Yes, indeed. Meta cannot see this.

Aha! I think I've got it:

As a result, I have no idea what you're talking about. No one mentioned multiplication, the issue was division. — Metaphysician Undercover

This rules out that you understand the language and refuse to speak it. You genuinely don't speak the language of math.

Okay, let me show you where the multiplication is. Let's revisit this:

(a) 1/9 can be one if the whole group is nine, (b) it can be two if the whole group is eighteen, (c) it can be three if the whole group is 27, (d) it can be four if the whole group is thirty six, and so on and so forth. — Metaphysician Undercover

(a) 1/9 of nine is $\displaystyle\frac{1}{9}\times 9 = 1$
(b) 1/9 of eighteen is $\displaystyle\frac{1}{9}\times 18 = 2$
(c) 1/9 of 27 is $\displaystyle\frac{1}{9}\times 27 = 3$
(d) 1/9 of thirty six is $\displaystyle\frac{1}{9}\times 36 = 4$
Do you see the multiplication now?

Since I think we've finally nailed down the problem, I'll keep to just the key parts.

If you think that the quantity represented by "one" can be divided in any way that you please, then you deny the meaning of "one" as a single thing, — Metaphysician Undercover

Two major problems with this MU:

• When I slice one pizza into eight slices, it's still one pizza. So no, I don't deny singularity by saying that something can be divided.
• Ignoring the fact that I can still say it's one pizza, I can slice a pizza into multiple parts anyway. If there's some sense in which that denies your concept of singularity, then your concept of singularity is broken, because those pizzas can be sliced

(Of course, you could always deny pizzas are real).

What you fail to understand, MU, is that many things can be divided, even if you count one of them. Also, lots of things have whole-part relations; given a loaf of sliced bread with 24 (equal) slices per loaf, I can give you 3 loaves, or 3 slices... I'm still doing nothing but counting, but I'm giving you different "particular quantities" of bread. The slice quantity is much smaller than the loaf quantity. This is what's known as a unit. If I give you 3 slices, I'm giving you 3/24 loaves. We might also say 3/24 of one loaf = 3 slices. We can also apply units to continuous measurements, such as lengths along those dimensions you alone denied exist.

On an interpersonal note:

It appears you just haven't taken the time to understand what I was saying. — Metaphysician Undercover

Sorry, I have no idea of what you're talking about again. I wish you could make a greater effort to make clear what you want to say. — Metaphysician Undercover

^^-- this makes you look irresponsible and lazy. You're blaming me for not understanding you, and blaming me again for you not understand me. This conveys the message that you think your time is extremely valuable and my time is worthless. That's... not great optics.

Also I think you're putting the cart before the horse. We don't start with some definition of "number" and then see which things satisfy that definition. Instead we have the mathematical terms $1$, $\frac{1}{9}$, $\sqrt{2}$, etc. which mathematicians place in sets that they decide to name "natural number", "rational number", "real number", etc. and then lexicographers try their best to come up with an adequate description of what the word "number" means when they write their dictionaries.

As The Princeton Companion to Mathematics says:

In the long run, this intellectual ferment led mathematicians to let go of the vague notion of “number” or “quantity” and to hold on, instead, to the more formal notion of an algebraic structure. Each of the number systems, in the end, is simply a set of entities on which we can do operations.

This standard "proof" is of course bullpucky. It's true, but not actually a proof at this level. Why? Well, as you yourself have pointed out, the field axioms for the real numbers say that if x and y are real numbers, then so is x+y. By induction we may show that any finite sum is defined. Infinite sums are not defined at all.

To define infinite sums, we do the following:

* We accept the axiom of infinity in ZF set theory, which says that there is an infinite set that models the Peano axioms. — fishfry

Wouldn't we have to do that to even be able to talk about 0.999...? Or can we somehow deal with "infinite" sequences without the axiom of infinity?

Also it seems to me that what you call "numbers" mathematicians call "natural numbers" (or maybe "integers"; do you consider negative numbers as numbers?). There's more than just natural numbers in mathematics; there's rational numbers that include the commensurable fractions like 1919, real numbers that include irrational numbers like 2–√2, and more. — Michael

I can't answer this question without an acceptable definition of "number". It seems mine has been rejected. But as I just described, one cannot be divided by nine, because it means that one is a multiplicity, when it is defined as a single, or simple.

don't see what purpose there is in saying that non-natural numbers aren't numbers, and latching onto the OP saying "as a matter of representing numbers" completely misses the point of this discussion. — Michael

The point is that a ratio is not a number, it is a relation between two numbers. And some ratios cannot be expressed as a number, the relation expressed by pi for example. Which ratios can and cannot be expressed as a number is a matter which might be discussed. I am taking the extreme position to claim that none of the following 1/2, 1/3, 1/4, 1/9, etc., can be expressed as a number, because it contradicts the definition of "one" to say that one is composed of a multiplicity which can be divided.

If the others, in this thread, insist that all ratios can be expressed as a number, then I want to see that definition of "number" which allows for this. Or is "number" just some meaningless word which mathematicians can use however they please, in any random way?

For example, if the quantity expressed by "one" can be divided however a mathematician wants to divide it, then it must be a multiplicity composed of an infinity of parts. But that contradicts its definition of "single", and makes "one" into a meaningless term. Having no restrictions derived from what it means to be "a single object", or some such thing, a person might refer to any multiplicity whatsoever as "one". And this leaves "one" as an absolutely nonsensical term.

This rules out that you understand the language and refuse to speak it. You genuinely don't speak the language of math. — InPitzotl

Of course. If you're just now noticing, I refuse to use that deceptive language, loaded with contradiction in its axioms.

(a) 1/9 of nine is 19×9=119×9=1
(b) 1/9 of eighteen is 19×18=219×18=2
(c) 1/9 of 27 is 19×27=319×27=3
(d) 1/9 of thirty six is 19×36=419×36=4
Do you see the multiplication now? — InPitzotl

That is just deception. To divide nine into nine parts, or to divide eighteen into nine parts is very clearly division. To express this as multiplying nine by another number, "1/9", or 18 by "1/9", is an act of deception. Claiming that division is multiplication is deception.

When I slice one pizza into eight slices, it's still one pizza. — InPitzotl

No it isn't it has been divided. Either it is one object, or it is eight objects. To claim both is to claim contradiction.

What you fail to understand, MU, is that many things can be divided, even if you count one of them. Also, lots of things have whole-part relations; given a loaf of sliced bread with 24 (equal) slices per loaf, I can give you 3 loaves, or 3 slices... I'm still doing nothing but counting, but I'm giving you different "particular quantities" of bread. The slice quantity is much smaller than the loaf quantity. This is what's known as a unit. If I give you 3 slices, I'm giving you 3/24 loaves. We might also say 3/24 of one loaf = 3 slices. We can also apply units to continuous measurements, such as lengths along those dimensions you alone denied exist. — InPitzotl

If this is true, then we need to define how to distinguish a whole from a part, so that we are not referring to the part as "one", when it is really 1/8 of the whole, and we are not referring to the whole as "eight" parts when it is really one whole. I would enter a discussion of parts and wholes with you, so long as we have principles whereby we can distinguish one from the other, and not just randomly decide to call this a part, and that a whole, at will throughout the discussion, because that would get nowhere.

Also I think you're putting the cart before the horse. We don't start with some definition of "number" and then see which things satisfy that definition. Instead we have the mathematical terms 11, 1919, 2–√2, etc. which mathematicians place in sets that they decide to name "natural number", "rational number", "real number", etc. and then lexicographers try their best to come up with an adequate description of what the word "number" means when they write their dictionaries. — Michael

That's putting the cart before the horse. Before deciding which items go into which set, we need to define the conditions of the set. No one puts a whole bunch of random terms into one set, then names the set "numbers". if that were the case, why wouldn't we put "house" and "car" into that set called "numbers" as well?

But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1 — fdrake

This still begs the question about the difference between a limit and an actual infinite sum. Your reasoning shows that you don't run out of natural numbers 'until' infinity.

That's putting the cart before the horse. Before deciding which items go into which set, we need to define the conditions of the set. No one puts a whole bunch of random terms into one set, then names the set "numbers". if that were the case, why wouldn't we put "house" and "car" into that set called "numbers" as well? — Metaphysician Undercover

They start by defining the following set:

N = {0, 1, 2, 3, ...}

Then they define the set "Z" as the set that contains the elements in set "N" and also their additive inverses, i.e. {-1, -2, -3, ...}.

Then they define set "Q" as the set that contains the elements a/b where a and b are elements in set "Z" and b is not 0.

Then they define set "R" as the set whose elements are the limit of a convergent sequence of the elements in set "Q".

Then they define set "C" as a + bi where a and b are elements in set "R" and i is a formal square root of −1.

(There are more formal ways of defining these sets than the above, but the above is easier to understand for someone who doesn't know much about maths. See here if you want something more exact).

None of this depends on there being a formal definition of the English language word "number" which is what I was talking about and which is where you're getting lost.

If you absolutely must have a formal definition of "number" then lets go with "any member of $C$" so we can get back on topic.

Of course. If you're just now noticing, I refuse to use that deceptive language, — Metaphysician Undercover

As a result, ->I have no idea what you're talking about<-. No one mentioned multiplication, the issue was division. — Metaphysician Undercover

^-- One of those two things is a lie. Most charitably, you're incapable of using the language.

Of course. If you're just now noticing, I refuse to use that deceptive language, — Metaphysician Undercover

^-- This is straight up paranoia. Deception has two parts... the advertised meaning, and the true meaning... the advertised meaning must be what you want to trick the other person to believe... the true meaning must be something different. We don't have that here... we only have one part... the usage.

The problem is that you don't understand the language, therefore, you spin this meaningless narrative that people are trying to deceive you (sprinkled with paranoia). It's meaningless, because there's nothing you can say that you're being deceived to believe... it's a working language, you just don't understand it. This is trivial to show in your debates; you don't even bother to debate what doesn't work. This is perceived on your part as non-compliance: "I refuse to use that deceptive language", but in terms of truth, that's hollow... your only possible genuine complaint is that the language doesn't work... to show it doesn't work you have to know how to speak the language, in order to construct the contradiction.

Otherwise, the only possible complaint you have left, the one you keep whining about, is that it's not the same language as your uninteresting one.

(e) To divide nine into nine parts, or (f) to divide eighteen into nine parts is very clearly division. — Metaphysician Undercover

Sure, but that's just division:
(e) $\displaystyle 9 \div 9 = 1$ <- "divide nine into nine parts"
(f) $\displaystyle 18 \div 9 = 2$ <- "divide eighteen into 9 parts"
...and this is using a fraction:
(g) $\displaystyle \frac{1}{9} \times 9 = 1$ <- " 1/9 of 9"
(h) $\displaystyle \frac{1}{9} \times 18 = 2$ <- "1/9 of 18".
Going back to our loaves:
(i) $\displaystyle 3 \div 24 = \frac{3}{24}$
There's no deception here, there's only confusion... on your part. Nobody who uses this language is confused. This is just how the language works. A ninth is the multiplicative inverse of nine. A twenty fourth is the multiplicative inverse of twenty four. Dividing by nine is equivalent to multiplying by a ninth. "A ninth of" is multiplying by a ninth; just as "five ninths of" is multiplying by five ninths. There's no problem here.

If this is true, then we need to define how to distinguish a whole from a part, so that we are not referring to the part as "one", when it is really 1/8 of the whole, and we are not referring to the whole as "eight" parts when it is really one whole. — Metaphysician Undercover

We do it by applying a unit. A slice is a part of a pizza. One pizza. Eight slices. It's so easy, everyone but you does it all the time!

I would enter a discussion of parts and wholes with you, so long as we have principles whereby we can distinguish one from the other, — Metaphysician Undercover

A yardstick measures 1 yard. It has 3 feet in it. Each feet has 12 inches. Those 12 inches usually are marked in fractions of an inch; typically at least an eighth of an inch. Now don't get scared... an eighth of an inch is part of an inch which is part of a foot which is part of a yard. Parts are transitive; an inch being part of a foot being part of a yard means an inch is part of a yard. We call the "whole" we're talking about a unit, and we just specify it... that's all there is to it. I say the yardstick is one yard long. That is three feet long, 36 inches long, and 288 eights of an inch long.

and not just randomly decide to call this a part, and that a whole,

Why not? That's how you use language. You have to specify the thing you're talking about, even if it's a part. I drive my car. I drive it into traffic. I turn the steering wheel. There's no problem doing this, outside of you having a problem with it, but that's not our problem. Let me rephrase this so that it sinks in:

If the only problem with the language is that you have a problem with it, then you are the problem.

Nobody has to talk to you before they use an IEEE-754 64-bit float in a program and, even if they do talk to you, you're giving them absolutely no reason to care. Let me rephrase that... nobody cares about your impoverished language. The reason I'm talking to you is that I care about you.

because that would get nowhere. — Metaphysician Undercover

I almost agree... your whining about something that works gets us nowhere. The only part where I disagree is that your whining about something that works has negative effects.

You don't seem to understand. "One" does not represent a quantity which can be divided. — Metaphysician Undercover

Curious about your 1/9 concerns. A while back you told me you believe in rationals but not sqrt(2). But now you don't seem to believe in rationals. What's up?

Secondly, can you give me a yes or no response to this question? Do you agree, either by personal understanding or by taking my word for it, that regardless of whether .999... = 1 is "true" in any metaphysical sense, it is still the case that it's a formal consequence of the axioms of ZF set theory?

Secondly, can you give me a yes or no response to this question? — fishfry

Still trying to nail jello to the wall?

Still trying to nail jello to the wall? — tim wood

@Metaphysician Undercover's in my head for sure. But I think it's a good clarifying question. Whether he accepts the formalism on its own terms even if not in any ultimate sense.

But I think it's a good clarifying question. — fishfry

...and here is the gift the crackpot gives to the world. Occasionally.

Exorcisms cheap. Or, go home and read my book, How to Keep One Step Ahead of Your Mind and you'll feel better in the morning. Or, best, three ounces or so of a decent single-malt Scotch.

...and here is the gift the crackpot gives to the world. Occasionally. — Banno

Exorcisms cheap. Or, go home and read my book, How to Keep One Step Ahead of Your Mind and you'll feel better in the morning. Or, best, three ounces or so of a decent single-malt Scotch. — tim wood

I'm a grizzled veteran of .999... threads. I was dismayed to find one had sprouted up here, but I'm powerless to resist. I've sworn off them many times without success.

I've sworn off them many times without success. — fishfry

With this, I have great sympathy. I compared it earlier to the compulsive tonguing of a broken tooth.