Is 1 = 0.999... ? Really?

Not at all. And note that this isn’t merely “by me”.

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

Lol

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

Keep going.

“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?

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)

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.

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.

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

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.

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.

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.

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”.

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

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.

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”.)

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!

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?)