Is 1 = 0.999... ? Really?

Alright.

[s]I’m inclined to think the word “endless” actually means “without an end”. If “mindless” means “without a mind”, it’s only sane to assume that “endless” means “without an end”. (And not, as you say, without a beginning.)

Google appears to be in agreement with me:

But there’s a bigger problem than that.

Even if the word “endless” means “without a beginning”, I don’t see a way to apply it to sets since the word “beginning”, just like the word “end”, is not defined with respect to sets.[/s]

I misread the above quote which is why the above is struck through.

The only part that does not need to be struck through is this last bit:

Take the set of ternary digits ({1, 2, 3}). What’s the beginning of it? And what’s the end of it? Note that sets have no order, so you can’t say the beginning of this set is (1) and the end of it is (3).

So if “infinite” means “endless”, and endless means either “without an end” or “without a beginning”, what does it mean to say that a set has no end/beginning?

It’s actually sequences, not sets, that possess the ability to have a beginning and an and.

The sequence ((1, 2, 3)) starts with (1) and ends with (3). It has a beginning (the first element) and an end (the last element.)

There are sequences that have a beginning but no end e.g. ((1, 2, 3, \dotso)). Such are necessarily infinite.

Then there are sequences that have an end but no beginning e.g. ((\dotso, 3, 2, 1)). Such are necessarily infinite just as well.

And then there are sequences that have no end and no beginning e.g. ((\dotso, -3, -2, -1, 0, +1, +2, +3, \dotso)). These are necessarily infinite just as well.

But there are ALSO infinite sequences that have a beginning and an end e.g. ((1, 3, 5, \dotso, 6, 4, 2)).

Depending on how you define the term “infinite”, it might be a contradiction to say that such sequences are “infinite”. But you can’t deny that they exhibt the same property that what we nowadays call “infinite sets” do – the number of their elements is greater than every integer.

But there’s an even bigger problem than this. You’re actually arguing that the word “infinite” means more than “endless”. You are actually saying that the word “infinite” means “a never-ending process of increase”.

The problem with such a claim is that neither sets nor sequences exist in time. They do not occupy time. Thus, they can’t be one thing at one point in time and another thing (or the same exact thing) at another point in time. They have no temporal existence. Since they have no temporal existence, they cannot change (since change is a difference between two points in time.) Since they cannot change, they cannot increase in size. Their size is fixed. The set of natural numbers (N = {1, 2, 3, \dotso}) isn’t ONE size at one point in time and ANOTHER size at another point in time.

Sequences and sets, being highly abstract concepts, have no temporal existence. But there are special (non-mathematical) kinds of sets and sequences that do have such a property.

For example, the set of apples on one’s table is something that exists through time. Another example is a train – a sequence of wagons. These things have temporal existence but they AREN’T mathematical entities. And this thread is ENTIRELY about mathematics. So there’s no point in discussing them (other than to distract.)

Yet another completely irrelevant concept (:
(Which also does not mean what you think it means.)

Magnus,

You’re playing with concepts in a very sloppy way.

Infinitesimals are surreal numbers or hyperreal numbers.

You think you can use a magic wand to make a number like:

0.333pi3333…

You can’t do that.

One of the big problems you face with your arguments about infinities is that you say that if they subtract, then they can also ADD (but you only bother with subtraction because you think that’s your strongest argument.).

If you stated, 2/3rds tacked on the end of 1/3, you’d be immediately laughed out of this thread!

0.333…0.666…

Everyone would call you stupid.

How about this?

0.333…9

Same thing!

There is actually no known math utility for hyperreals, but you believe them because you can make the sentence! There are an infinite number of sentences that aren’t true:

“Existence doesn’t exist!”

There, I made a sentence! Is it true just because I made it?

Of course not!

You’re writing math that is false (hyperreals)

Is the entire point of your presence here in this thread to delcare that you’re right and that everyone else is wrong?

Sort of like what you’re doing here?

I am still waiting for you to begin addressing my points.

Oh, are we projecting now Magnus!?

You wrote an entire message about hyperreals and I responded to it!

Magnus, you’re talking over people, you aren’t talking to them.

Is

0.333pi333…

A number or not a number?

Sure, you can call me crazy for other posts in other threads (that’s what you’ve reduced yourself to), why don’t you try sticking to this thread for a change?

You know why you can’t ? Because you feel the screws of disproof slowly twisting at your skull in this thread and you don’t know what else to do.

Well, I’m still waiting for you to begin addressing my points.

I wrote a message, yes, but I am not really sure it was a message about hyperreals.

Are you sure? Can you check one more time?

You didn’t directly respond to any of the points I raised.

  1. I said that mathematical sets have no notion of beginning and no notion of end which means that your definition of infinity cannot be applied to them.

You said NOTHING in response to this.

  1. I said that it’s sequences, and not sets, that have the ability to possess a beginning and an end.

You said NOTHING in response to this.

  1. I said that sequences that have an end but no beginning are infinite (that they are infinite even though they have no end because the number of elements they consist of is larger than every integer.)

You said NOTHING in response to this.

  1. I said that sequences that have BOTH a beginning and an end are infinite (that they are infinite even though they are bounded from both sides because the number of elements they consist of is larger than every integer.)

You said NOTHING in response to this.

  1. I said that mathematical sets and sequences do not exist in time which implies they aren’t processes (since processes exist in time.) (And since an infinite set, being a set, is not a process, it also follows that it is not a never-ending process of increase.)

You said NOTHING in response to this.

  1. I said that non-mathematical sets and sequences that do exist in time are of no relevance to this topic. (Because this topic is entirely about mathematics and because mathematical entities do not exist in time.)

You said NOTHING in response to this.

And how about things I didn’t say but should have said? For example, the fact that your definition of infinity as that which is endless does not imply that infinity is a never-ending process (something you insist the word “infinity” implies.)

Perhaps you need to learn how to hold a conversation first?

So instead of directly responding to what I said, you responded to something you imagined . . . infinitesimals, hyperreals, numbers such as 0.333pi333… which I have no idea what they represent and so on.

It’s not a number because Pi is not a digit. I don’t see how this thing you came up with out of nowhere is of any relevance whatsoever. Care to explain?

Of course pi is not a digit. Pi is an algorithm. It’s a process that never ends!

Like I stated before:

In infinity, if you have a series that is every other number and use this as proof that Infinite series have different sizes!

All I have to do is write an algorithm that they all take a step back or forward like this:

If the first number doesn’t correspond then you either send all the other numbers up one step or you send all the numbers back one step.

With infinities, either way you choose will still
Have everyone holding everyone’s hand.

With finiteness, this is not possible …

Magnus,

You are treating infinite as finite!

I am waiting for you to do two things:

  1. address at least one of my points

  2. stop abusing “Quote” button

Here’s a song to that end:

[youtube]https://www.youtube.com/watch?v=dJxHDpKZxOs[/youtube]

Magnus,

Your last two post haven’t been discussions, they’ve been you saying, “I lost the debate and I don’t know what else to do”

You have to respond to the points that I raised.

I’ll grant you ignorance here.

In discussions on Internet forums, it’s on you to bring up AGAIN!!! that which you think I’m ignoring!!!

If you’re really lazy, just make a link to the post you think I ignored! And tell me how I ignored it

Pi is neither an algorithm nor a process. (The word “algorithm” and the word “process” mean two different things. Process is what you get when you execute an algorithm. Algorithms themselves aren’t processes. They are merely descriptions of processes.)

Most importantly, even if Pi is a terminating decimal, it still wouldn’t be a digit.

Well, I already did. Just a couple of posts ago. You even quoted it. Here it is. I made (6) different claims and you have yet to respond to a single one.

I read the first sentence, and holy shit! I missed that post! Sorry Magnus ! I’ll read it now! Our last four posts have been a misunderstanding

How does one miss a post that they quoted and responded to? (:

Ok Magnus, I read your post.

I answered every point you made before you made that post.

In short, to all of it, if you believe infinities have beginnings and ends (both), then we have no discussion or conversation.

It’s implied that the endlessness of infinity means that infinity is not an object. It can only be described as a process.

You also completely ignored my point that in infinite sets, if you choose every other number as “non-correspondent” that you can use the simple algorithm of taking 1,2,3 steps forwards or backwards and EVERYONE will still be holding hands again, which is impossible in finite sets.

How about you discuss THAT! Instead of ignoring it for 3 pages!!

Care to show exactly where?

I am curious to see if what you’re saying here (that you responded to each one of my points before I raised them) is true.

I didn’t say that “infinities” can have beginnings and ends. I said very clearly that it is infinite sequences that can have beginnings and ends. And I further explained that this is because the word “infinite” simply means “a number larger than every integer”. So when we say that a sequence is infinite, we’re not actually saying that it’s without an end (where by “end” I mean “the last element in the sequence”), but rather, that the number of its elements is larger than every integer. This is exactly the same thing we do with sets. When we say that a set is infinite, we are not saying that it is without an end (because the word “end” is not defined with respect to sets) but that the number of its elements is larger than every integer.

I am pretty sure you did not directly respond to the following point (point #1) anywhere in this thread:

Take any finite set of your choice. You can take the set of ternary digits, for example.

({1, 2, 3})

What does it mean to say that this set has an end?
(You are NOT allowed to answer with “It means the number of its elements is an integer”.)

What’s the last element in this set?
There is no such thing because sets have no order.

How does the word “endless”, which means no more than “without an end”, imply a never-ending process of increase?

I responded to that several times in the past.

The problem with that claim of yours is that it contradicts your earlier statements.

You earlier statements imply there’s no one-to-one correspondence between the two sets.

Take a look here.

Magnus,

We have two separate debates in this thread.

0.999=1 (or not)
Orders of infinity exist (or not)

I’m going to resolve both in this post.

9/10+9/100+9/1000… does equal 1 if you add

1/10+(0/10) + 1/100+ (0/100) + 1/1000 etc…

To make the argument that 0.999…=1, you have to argue that addition and non operators are EQUAL!!!

A contradiction! So either addition means something, or all numbers are equal to each other without even using operators.

I’m using (to make this proof) an expanding algorithm, as long as it keeps moving (and carrying), 0.9 + 0.1 always equals 1 and 0.99 + 0.01 always equals 1!

Have you personally EVER seen an infinite sequence that’s not an algorithm? Have you EVER seen a bound infinity all at once?

Nobody ever has nor ever will. Because infinity just keeps going and going and going. By definition!

That means that no matter how far you look, it just keeps going, because the algorithm doesn’t terminate! in the mathematically ideal sense of what infinite expansion or infinitesimals are defined as.

Thus: it’s a process.

In other words, you are going to ignore everything I said and simply present a new argument of yours.

That’s not a good way of making friends.

It has nothing to do with seeing and everything to do with defining.

The term “infinite sequence” does not refer to an algorithm. (And it also does not refer to a process.)

The only POSSIBLE way that your sentence is true is if you’ve seen ALL the members of an infinite sequence in your head!! If you did or do that, then yes, infinity is not an algorithm or process.

That’s quite a claim Magnus … that you hold every number in your head!!!

Actually, it’s laughable!

An algorithm is a finite sequence of instructions on how to perform a task.

A process is what you get when you run, execute or follow an algorithm.
(Note that the two words aren’t synonymous. Not only that but algorithms aren’t even processes.)

An infinite sequence is not an algorithm for two reasons:

  1. because algorithms are FINITE sequences (whereas infinite sequences are INFINITE sequences)

  2. because algorithms are sequences of SPECIFIC THINGS (namely, instructions on how to perform a certain task) (whereas infinite sequences can be sequences of pretty much anything e.g. animals)

It has ABSOLUTELY NOTHING to do with whether one can hold all of the members of an infinite sequence in one’s head or not.