Is 1 = 0.999... ? Really?

What does it mean to say that someone can hold something in their head?

Does it mean that someone’s head is big enough for that thing to fit inside?

In other words, does it mean that the volume of one’s brain is equal to or greater than the volume of that thing?

If this is the case, what’s the volume of the infinite sequence that is ((1, 2, 3, \dotso))?

If the volume of every number in that sequence is exactly one infinitesimal (cm^3), then the total volume of that sequence (assuming that only its elements have volume) is exactly (1cm^3) (since infinity times infinitesimal is (1)) which makes the volume of that infinite sequence smaller than every human brain.

You were asked to clarify, not to repeat what you said.

That’s what communication is about.

On the other hand, the fact that something is larger than one’s head does not mean that it is a process. There are many things that are larger than our heads e.g. airplanes. Does not mean that airplanes are processes.

I like that analogy!

You can hold a plane in your “minds eye”, you cannot hold 0.999… in your “minds eye”

Does that help?

These are very abstract metaphors here!

In order to be able to see or visualize anything, one needs time. If you’re sitting in front of a TV and I’m flashing a picture on it (let’s say the picture is shown for no longer than (1) millisecond), you wouldn’t be able to tell me anything about it. The more you want to see or visualize something, the more time you need. If it takes you one second to visualize a single apple, it will take you an infinite number of seconds to visualize an infinite number of apples. If it takes you one infinitesimal second to visualize a single apple, it will take you a second to visualize an infinite number of apples.

Most importantly, the fact that we can’t see or visualize something within a finite period of time does not mean that that thing is a process.

Ahh… that’s an interesting argument. You gave me something new to work with.

If infinity is not a process(as I say it is), then the odds of an infinitesimal (say us in an infinite universe) are exactly zero percent! Actually, and I say this to the best mathematicians on earth (and hopefully you can understand it magnus!) if there are always infinitely larger infinities, and our cosmos is finite (or a lesser order of the highest infinity), than at the highest cardinality, there is a zero percent chance for our entire universe to exist!

If however, hyperreals can’t exist, then our odds of existing are not zero percent, but rather 100%.

I figured out a way to put this in layman’s terms…

Let’s say that only one penny exists in all of existence, but you have an infinite number of dimes (they never stop) - your odds of finding the penny are zero percent! As if it didn’t and never did exist.

I’d say the probability of finding a penny is (\frac{1}{\infty + 1}). That’s close to (0) but it’s not (0).

The question I want to ask is:

How does this prove that the word “infinity” refers to a process?

That’s a good question!

We have inferential proof.

We can inferentially prove the sequence 1,2,3,4,5,6,7. Etc… not only goes on forever (without repeat), but always increases in size.

We can’t actually “see” in our “minds eye” ALL of it!!

ALL is not a “word that infinity understands”. These are metaphors to be sure!

All assumes completion, such as the number 1, it’s abstracted as an “object” (again, a type of metaphor)…

But! Is infinity (or an infinite sequence) an object like the number 1 is?

The answer is a resounding “no”. Because it’s not an object, it’s a verb, it’s action itself.

What is inferential proof?

I can agree that the sequence ((1, 2, 3, \dotso)) has no end and that it contains no repetitions. However, I cannot agree that it increases in size.

Can you provide a syllogism?

If you cannot expose your reasoning, all we can do is tackle your claims by providing counter-arguments i.e. arguments that prove the opposite of what you say.

This means that, unless you expose your reasoning, all I can do is say that the term “infinite sequence” does not refer to something that exists in time which means that it cannot increase in size and that it’s not a process in the first place. I can back this up by quoting Wikipedia. But I’ve already done most of the work, so the only thing that is left right now is to ask you to expose your reasoning. Such a feat requires a degree of self-consciousness and I’m not really sure you can pull it off but one can always hope.

Magnus,

I already did.

No possible being can hold “all” of an infinite sequence in their “minds eye” at once, all they can do is to infer it.

An object has an end. Say, a couch. Thus we call it a couch. Very simple. An infinite number of couches?

No way! We can infer it, but we cannot count it.

Here’s a flowchart demonstrating a structured approach to discussing ideas, one that is conducive to resolving disagreements. (This is no quick fix, so no promises of the form “Make everyone agree within a record period of time!” are made. My sole claim is that it’s an approach that is better than other approaches. If it takes more than (1,000) pages for people to come to agreement using this approach, it merely means that other approaches would have taken much longer than that.)

Ecmandu loves algorithms, so he should be able to appreciate this flowchart:
How to Discuss Ideas

It’s a pretty simple flowchart, actually.

So when I say that someone did not address a claim I made, I am simply saying that they did not follow the steps outlined in this algorithm.

The key part is the idea that the right way to respond to mere assertions is different from the right way to respond to arguments.

For example, if I make a statement that (0.9 \neq 1), without explaining how I arrived at such a conclusion, the adequate way to respond to it, in case you disagree with it, is by offering a counter-argument i.e. by providing an argument that proves that (0.9 = 1). For example:

(3 \times \frac{1}{3} = 1)
(\frac{1}{3} = 0.333\dots)
(3 \times \frac{1}{3} = 3 \times 0.333\dotso = 0.999\dotso)
Therefore, (1 = 0.999\dotso).

Whether or not the argument is sound, whether or not its conclusion is true, whether or not the person understood the original claim and whether or not the person presenting such an argument is stupid or smart are completely irrelevant. The response is an adequate one for the simple reason that it provides a counter-argument to a previously made claim.

However, that wouldn’t be a proper response in the case I offered the following argument . . .

1. (1) is a number
2. (0.999\dotso) is not a number because it does not end
3. A number cannot be equal to something that is not a number
4. Therefore, (1) is not equal to (0.999\dotso)

Here, you have to show what’s WRONG with the presented argument and not merely argue against the conclusion. You have to explain why you think the argument is UNSOUND. Is it logically invalid? If so, why? Are there any premises that are false? If so, which ones and why?

But if all you do is COMPLAIN about how other people are not up to your expectations, then you will get nowhere. You will waste your time and even achieve the opposite of what you wanted to achieve in the first place.

Magnus,

Stop projecting man! Fuck! Even I still do it from time to time! It’s annoying as fuck!

You (in stating) that 0.999… is not a number is saying the exact same thing I’m saying when I state that infinities are not objects.

We’re just using different terminology for the same conclusion.

If “all” doesn’t ever describe infinity, then infinity ceases to possibly be a noun, it forces it to be a verb.

That’s not my statement.

Ok. You copied someone else’s statement. Prove that it’s wrong.

I don’t see a syllogism here. I can’t see it in a previous post either.

Alright.

Let’s say the full argument of that imaginary person of mine goes something like this:

1. (0.999\dotso) has no end
2. Numbers must have an end
3. Therefore, (0.999\dotso) is a number

I disagree with the second premise. The word “end” is not defined with respect to numbers. What does it mean for a number to have an end or to have no end?

The first premise is stating that the infinite expression represented by (0.9 + 0.09 + 0.009 + \cdots) has no end. I agree with that. However, that infinite expression is a symbol, it is not the symbolized. It is that which represents, not that which is represented. It says NOTHING about that which is represented. So even if we accepted the second premise (that numbers must have an end), the conclusion does not follow.

Man you’re an asshole Magnus!

You have any idea how hard it is to write a syllogism?

You know what I’m saying, and you know what it means, and still you want a syllogism from ME!

Why don’t you write the fucking syllogism since in your other thread (in rant about the purpose of these boards) you criticized people for how lazy they are, and only non-lazy people are the only worthwhile beings - in other words walk that talk.

Magnus!

You are so confused in this message!

Of course 0.999… is the symbol and not the symbolized. I’ve been saying that the whole fucking time!

The symbolized NEVER ends!!! NEVER is a temporal word!

I don’t understand the process of your reasoning which is why I am asking you to present a syllogism.

If it’s too hard for you to write a syllogism, you have nothing to do on a philosophy forum.

My position is that it’s the symbol, and not the symbolized, that never ends.

And when I say that it’s the symbol that never ends, I do not mean to say that it’s the symbol (0.999…) that does so. That symbol is a finite sequence of characters, so it does end. It’s this other symbol that does not end – the one that cannot fit inside a post (because posts are finite.) The “invisible” one, so to speak.

Let me try to explain this with a different number. Consider (1.000\dotso). This is a finite symbol because it is a finite sequence of characters. It represents (1). I am pretty sure you agree. This symbol, however, is a shorter version of another symbol – the infinite one – that also represents (1) despite the fact that it is infinite. It’s a symbol best captured by the sentence “A one, followed by a dot, followed by an infinite number of zeroes”. That thing is a symbol, it’s not the symbolized. The symbolized is a number – specifically, it is number (1) – and numbers have no notion of end.

What does it mean to say that a number has an end or that it does not have an end?