Is 1 = 0.999... ? Really?

[Ecmandu]

Magnus, a paradox doesn’t mean that you’re right or wrong ... have you disproved the paradox?

No.

So let’s look at the two gods argument again. A god is exactly half as omniscient as another god. Can that occur?

What does it mean to know half of everything if everything is infinite?

What about everything * 2?

Do you see why so many of us are debating you yet???

What’s everything * 2?

You mention contradictions frequently, but have an absolute blind spot to this very basic contradiction.

That’s why everyone else here is arguing against you.

[Magnus Anderson]

You are, once again, dismissing counter-examples.

That's what you're doing but you're not recognizing it.

Allow me to repeat myself:

By definition, if you have a bunch of things in one place (regardless of their number) and you subtract one thing, you get a smaller number of things. To say otherwise is to say that you didn't really perform the operation of subtraction, which is a logical contradiction.

Whatever the quantity, if you subtract one from it, the quantity must change.

Suppose there is an infinite line of people somewhere in the universe and that YOU are one of the people waiting in it.

Suppose now that I take you by your hand, remove you from the line and place you somewhere outside of it.

The line is the same as it was before except that you're no longer part of it. Noone joined the line, noone left it -- except for you.

If you say that the number of people waiting in that line is the same as before, it either means that I didn't really took you out of that line (that you're still there) or that I did but that someone else joined it. Both are contradictions.

You never addressed this argument.

[Ecmandu]

You are, once again, dismissing counter-examples.

That's what you're doing but you're not recognizing it.

Allow me to repeat myself:

By definition, if you have a bunch of things in one place (regardless of their number) and you subtract one thing, you get a smaller number of things. To say otherwise is to say that you didn't really perform the operation of subtraction, which is a logical contradiction.

Whatever the quantity, if you subtract one from it, the quantity must change.

Suppose there is an infinite line of people somewhere in the universe and that YOU are one of the people waiting in it.

Suppose now that I take you by your hand, remove you from the line and place you somewhere outside of it.

The line is the same as it was before except that you're no longer part of it. Noone joined the line, noone left it -- except for you.

If you say that the number of people waiting in that line is the same as before, it either means that I didn't really took you out of that line (that you're still there) or that I did but that someone else joined it. Both are contradictions.

You never addressed this argument.

Magnus! If I subtracted you, you’d cease to exist. You would not be part of “another set”. You’d have no bearing on the infinity.

[Magnus Anderson]

What does it mean to know half of everything if everything is infinite?

What about everything * 2?

Do you see why so many of us are debating you yet???

What’s everything * 2?

You mention contradictions frequently, but have an absolute blind spot to this very basic contradiction.

That’s why everyone else here is arguing against you.

You aren't following, Ecmandu.

I've addressed this argument before. Basically, the word "infinite" does not mean "the largest number possible" (let alone "everything".) The word "infinite" means "endless".

[Ecmandu]

What does it mean to know half of everything if everything is infinite?

What about everything * 2?

Do you see why so many of us are debating you yet???

What’s everything * 2?

You mention contradictions frequently, but have an absolute blind spot to this very basic contradiction.

That’s why everyone else here is arguing against you.

You aren't following, Ecmandu.

I've addressed this argument before. Basically, the word "infinite" does not mean "the largest number possible" (let alone "everything".) The word "infinite" means "endless".

So “everything” by you! Doesn’t mean “endlesss”?

Keep going! I’m curious.

[Magnus Anderson]

So “everything” by you! Doesn’t mean “endlesss”?

Not at all. And note that this isn't merely "by me".

[Magnus Anderson]

If I subtracted you, you’d cease to exist. You would not be part of “another set”.

To subtract does not mean to remove from existence. It means to remove from something where something can literally be anything.

[MagsJ]

Actually I've known all this math since over half my life ago, and I have been regularly keeping up with advanced level maths to this day for fun - I actually really enjoy it.

And still can't get it right. I'm impressed.

Lol

[Ecmandu]

So “everything” by you! Doesn’t mean “endlesss”?

Not at all. And note that this isn't merely "by me".

So, you think that endless doesn’t imply everything, and that everything doesn’t imply endless.

Keep going.

[Magnus Anderson]

"Everything" means "every element of some set" where "some set" can be literally any set.

E.g. "Everything you want" means "Every element of the set of things that you want".

The set can be finite or infinite. It does not matter.

"Infinite" merely means "without an end".

You can have a set made out of five infinite sets and the entire set would be more than any one of its five infinite sets.

You could ask "Which one of the sets do you want, Max?" And I could say "I want everything". That would mean "Give me all five infinite sets, Sir". In that particular context, "everything" would be representing five infinite sets. A finite number of sets each containing an infinite number of things.

[Ecmandu]

"Everything" means "every element of some set" where "some set" can be literally any set.

E.g. "Everything you want" means "Every element of the set of things that you want".

The set can be finite or infinite. It does not matter.

"Infinite" merely means "without an end".

You can have a set made out of five infinite sets and the entire set would be more than any one of its five infinite sets.

You could ask "Which one of the sets do you want, Max?" And I could say "I want everything". That would mean "Give me all five infinite sets, Sir". In that particular context, "everything" would be representing five infinite sets. A finite number of sets each containing an infinite number of things.

Ok, that’s interesting.

So what if I say, “everything that exists throughout all existence, the cosmos”

And then a guy like you comes along and says, “every Tickle me Elmo that exists throughout existence an all the cosmos”

They are not equalities are they?

So why try to put them forward as such?

[obsrvr524]

Especially if the universe is infinite (as it logically seems to be), there could be (and in fact logically seems to be) an infinity of apples, an infinity of oranges, and an infinity of pears.

The set of all three infinite sets is obviously greater than any one infinite subset.

[Ecmandu]

Especially if the universe is infinite (as it logically seems to be), there could be (and in fact logically seems to be) an infinity of apples, an infinity of oranges, and an infinity of pears.

The set of all three infinite sets is obviously greater than any one infinite subset.

Just basically speaking, this means an infinity of the finite. (The concept orange is finite) (Contradiction) or as mathematicians put it, “bound infinities”.

How do you bound an infinity (boundless)? (Contradiction)

[obsrvr524]

Just basically speaking, this means an infinity of the finite. (The concept orange is finite) (Contradiction) or as mathematicians put it, “bound infinities”.

How do you bound an infinity (boundless)? (Contradiction)

Were you being sarcastic? Was that intentionally incoherent? Or??

"infinity of the finite"
"concept orange is finite"
"bound infinities"

None of those seem to have anything to do with the discussion.

[gib]

Let's try this: imagine the scenario I described earlier, the one with two infinite parallel lines. For all intents and purposes, the same length. Now remove every odd point from one of the lines. Then move all remaining point into the spots left behind by the points you removed. According to you, the line with the points removed is now "shorter". But since we moved all points into the spots left empty from the points we removed, the lines are perfectly identical. It's as if we didn't remove any points at all. We're back to the initial state of the scenario. So here's your chance to shine. Help me understand how the lines are different now. Help me understand what it means that the line we removed points from is now "shorter".

So we have two infinite parallel lines that are of the same length. We pick one of them and remove every odd inch from it. Then we fill in the gaps that we created using remaining inches. By doing this, the gaps disappear leaving the two lines looking perfectly identical.

The problem is that there isn't enough inches remaining to fill in the gaps without creating new gaps elsewhere. This illusion is created by moving the gaps out of our sight.

If you don't see it, it's not there.

And if you keep pushing things out of your sight, you can keep reassuring yourself they don't exist.
Especially if this process is an infinite one (:

Here's the line we started with:
\( \bullet \bullet \bullet \bullet \bullet \bullet \cdots \)

Now here's the line with odd inches removed from it:
\( \circ \bullet \circ \bullet \circ \bullet \cdots \)

There's an infinite number of inches out of our sight. We don't see them, we merely see the ellipsis "..." which tells us there is more to this line than what we see. What we want to do now is take three inches from the remaining inches that we don't see, so that we can fill in the gaps. We can do that, because there's still an infinite number of inches remaining, so we do that and we get:

\( \bullet \bullet \bullet \bullet \bullet \bullet \cdots \)

Voila! The line looks like the original one! They now appear to be identical! But what happened to those gaps? Where did they go? Well, they went out of our sight. They didn't magically vanish. We don't know exactly where they went, but they are somewhere out of our sight.

So the lines aren't really identical. They merely look like they are.

The gaps can't magically vanish. The only thing we can do is push them out of our sight forever thereby creating an illusion that the two lines are identical.

They are not.

This "paradox" is known as Hilbert's paradox of the Grand Hotel:
https://en.wikipedia.org/wiki/Hilbert%2 ... rand_Hotel

I think Carleas mentioned it somewhere at the beginning of this thread (40-50 pages ago . . .)

I don't get the gist of your argument. Are you saying there aren't enough points to fill in all the gaps (I said points, not inches, but...), or that as soon as the gaps are out of sight, they stop being replaced?

If not enough points, how do you have not enough with an infinite amount?

Are you saying that we eventually run out of points to fill the gaps, and after the last point there's nothing but gap? <-- That would imply there's an end to the series of point, and you know how that argument goes.

[Ecmandu]

Just basically speaking, this means an infinity of the finite. (The concept orange is finite) (Contradiction) or as mathematicians put it, “bound infinities”.

How do you bound an infinity (boundless)? (Contradiction)

Were you being sarcastic? Was that intentionally incoherent? Or??

"infinity of the finite"
"concept orange is finite"
"bound infinities"

None of those seem to have anything to do with the discussion.

Magnus: “the concept of an orange is finite”

So what does infinite orange mean? Are you trying to tell me that you can have infinite oranges, but not single infinite orange? Well... if you can’t have a single orange and all those infinite oranges are equal as infinite orange, how can you have infinite oranges?

I’ve never joked, let alone joked sarcastically on ILP, even though in real life I do both abundantly.

Bound infinity has EVERYTHING to do with the discussion... it means infinity is a quantity

[Magnus Anderson]

I don't get the gist of your argument. Are you saying there aren't enough points to fill in all the gaps (I said points, not inches, but...), or that as soon as the gaps are out of sight, they stop being replaced?

I'm saying there aren't enough points. There aren't enough points to fill the gaps that are within our sight without creating gaps out of our sight.

Are you saying that we eventually run out of points to fill the gaps, and after the last point there's nothing but gap? <-- That would imply there's an end to the series of point, and you know how that argument goes.

I am not.

We always have enough points to fill the gaps that are within our sight. But each time we fill the gaps that are within our sight, we create new gaps out of our sight.

Consider that in order to fill a gap, you have to remove a point elsewhere; and that when you remove a point, you create a gap in its place.

Here's the infinite line with odd inches taken out:

\( \circ \bullet \circ \bullet \circ \bullet \cdots \)

Suppose you want to fill the first gap. How do you achieve that? By choosing an existing inch and moving it from its current place to the beginning of the line. You can pick any inch you want. There's an infinite number of them. You can pick the first inch in the line. Let us do so. We pick the first inch in the line and move it to the beginning of the line. By doing so, we fill a gap but we also create a new gap. This is what follows:

\( \bullet \circ \circ \bullet \circ \bullet \cdots \)

We don't get \( \bullet \bullet \circ \bullet \circ \bullet \cdots \). That would be creating new inches out of nowhere.

The interesting part is that you don't have to pick an inch that is within your sight. You can pick an inch that is outside of your sight. You can pick the 100th inch or the 1,000th inch or the 1,000,000th one. In each case, you'd be creating a gap in its place. But because it's out of your sight, it's convenient to ignore it and pretend that the line no longer has any gaps.

It's a trick. Something a magician would do. It's definitely not logic.

[Ecmandu]

Again Magnus,

Nobody has proven that there’s not a scatter set for the reals. Using your line logic, we’d have never ordered the rationals (the same logic applies)

In the absence of a scatter set, you can use my cheat.

1.) rational number
2.) uncounted number
3.) different rational number
4.) different uncounted number

Etc...

Even if we do prove that there’s no scatter set for the reals.... you can still use my cheat.

[gib]

I don't get the gist of your argument. Are you saying there aren't enough points to fill in all the gaps (I said points, not inches, but...), or that as soon as the gaps are out of sight, they stop being replaced?

I'm saying there aren't enough points. There aren't enough points to fill the gaps that are within our sight without creating gaps out of our sight.

Are you saying that we eventually run out of points to fill the gaps, and after the last point there's nothing but gap? <-- That would imply there's an end to the series of point, and you know how that argument goes.

I am not.

We always have enough points to fill the gaps that are within our sight. But each time we fill the gaps that are within our sight, we create new gaps out of our sight.

Consider that in order to fill a gap, you have to remove a point elsewhere; and that when you remove a point, you create a gap in its place.

Here's the infinite line with odd inches taken out:

\( \circ \bullet \circ \bullet \circ \bullet \cdots \)

Suppose you want to fill the first gap. How do you achieve that? By choosing an existing inch and moving it from its current place to the beginning of the line. You can pick any inch you want. There's an infinite number of them. You can pick the first inch in the line. Let us do so. We pick the first inch in the line and move it to the beginning of the line. By doing so, we fill a gap but we also create a new gap. This is what follows:

\( \bullet \circ \circ \bullet \circ \bullet \cdots \)

We don't get \( \bullet \bullet \circ \bullet \circ \bullet \cdots \). That would be creating new inches out of nowhere.

The interesting part is that you don't have to pick an inch that is within your sight. You can pick an inch that is outside of your sight. You can pick the 100th inch or the 1,000th inch or the 1,000,000th one. In each case, you'd be creating a gap in its place. But because it's out of your sight, it's convenient to ignore it and pretend that the line no longer has any gaps.

It's a trick. Something a magician would do. It's definitely not logic.

Ok, I see what you mean. When you move the first point to fill the first gap, you get a gap after that point two points long. Then to fill that gap, you need to move the next two points, which leaves a gap four points long. It seems that as you move the points down, you get an ever grow gap moving in the opposite direction. That indeed brings into question what the line ends up looking like at the end. Do you really get an identical line, point for point, or do you get a line with an infinitely long gap at the other end (somehow still an infinite number of points away)?

This problem arises when you imagine each point taking its turn to fill the gaps. If each point takes its turn, you'd need an eternity to complete the thought experiment and answer the question above. But what about each point moving at the same time? This is how we are to imagine Hilbert's Hotel. Each guest moves to the next room simultaneously, not one after the other. Of course, in the case of the gaps in the line, each point would have to move a different amount. The first point moves one position, the second point moves two positions, the third point moves three positions, etc..

In either case, I'm still not clear on how we're defining "longer" and "shorter".

[obsrvr524]

I suspect that asking a dog to distinguish green from red is a futile endeavor (paracingular sulcus).

[Ecmandu]

Also Magnus,

I thought about your utility reply more.

You say that

A.) you want to get to the truth
B.) there’s more utility (if utility means truth) to 0.9... equaling 1

This can be resolved with second grade math:

Rounding.

Rounding is very useful, but we know it’s truth is only in its utility, not because it is actually an equality.

That’s why we see so many “magic numbers” appear, because people are rounding digits, not proving equalities.

[gib]

Come to think of it, I'm not sure two scenarios are any different: two identical lines or one line with an infinite gap an infinite distance away. To say the gap is an infinite distance away is equivalent to saying it's at the end of the line. But then what's at the end of the other line? More line? For all intents and purposes, if we're talking about "the end of the line", I'd say it's fair to say that's where the lines end. So the gap has effectively been push out of the line and the two are once again identical.

(I realize the silliness of talking about the "end" of an infinite line, but I'm just following the logic that would come out of granting talk about a gap at the "end of the line".)

[Aegean]

Still debating the theoretical meaning of '1'?

Existence happens between the theoretical absolutes of 1/0.
There is no 'one' in reality - nor a nil. It is a mental abstraction that can refer to anything the mind detaches from space/time and places within vague space/time borders.
'One' is an idea, representing an arbitrary moment/place.
Like all ideas it can be defined by the mind - synthesized, manipulated, redefined and redefined, combined, in ways that go beyond the real.

It is a linguistic representation of a mental abstraction, created by the translation of sensual stimuli.

0.9999 is not one...it is a movement towards an absolute that does not exist, and therefore can never be attained.
It represents the fluidity of existence, in relation to the mind's abstraction.

[Ecmandu]

Still debating the theoretical meaning of '1'?

Existence happens between the theoretical absolutes of 1/0.
There is no 'one' in reality - nor a nil. It is a mental abstraction that can refer to anything the mind detaches from space/time and places within vague space/time borders.
'One' is an idea, representing an arbitrary moment/place.
Like all ideas it can be defined by the mind - synthesized, manipulated, redefined and redefined, combined, in ways that go beyond the real.

It is a linguistic representation of a mental abstraction, created by the translation of sensual stimuli.

0.9999 is not one...it is a movement towards an absolute that does not exist, and therefore can never be attained.
It represents the fluidity of existence, in relation to the mind's abstraction.

No 1 in reality eh?

I’m replying to you right now. You are a 1.

Nice try though!

[gib]

Still debating the theoretical meaning of '1'?

Existence happens between the theoretical absolutes of 1/0.
There is no 'one' in reality - nor a nil. It is a mental abstraction that can refer to anything the mind detaches from space/time and places within vague space/time borders.
'One' is an idea, representing an arbitrary moment/place.
Like all ideas it can be defined by the mind - synthesized, manipulated, redefined and redefined, combined, in ways that go beyond the real.

It is a linguistic representation of a mental abstraction, created by the translation of sensual stimuli.

0.9999 is not one...it is a movement towards an absolute that does not exist, and therefore can never be attained.
It represents the fluidity of existence, in relation to the mind's abstraction.

Why does everyone think that 0.999... has to "build up"? Like it's a process that needs time to complete?

If I say, "suppose you had a queue of 50 people," do you say, "wait... okay, the 50th person has just been added. Now we can talk about it." What if I say, "suppose you had a queue of 100 people"? Do we have to wait twice as long before continuing with the conversation?

0.999... is simply notation. It just stands for the idea of an infinite number of 9s. You're supposed to imagine you already have an infinite number of 9s.

(And really, it doesn't even represent that; it represents a quantity; the debate in this thread is: what is that quantity? Is it 1 or the next number before 1?)