Is 1 = 0.999... ? Really?

That’s not true. What I’m doing is I’m treating infinite quantities as if they are quantities.

Maybe you should start with the word “definable”. What does it mean? What does it mean to say that something is not definable?

In most cases, there is absolutely no need for symbols to look like what they represent.

The sentence “infinite line of green apples” looks nothing like the infinite line of green apples and it doesn’t have to. (Indeed, it would be a problem if it looked like the infinite line of green apples.)

If the purpose of symbols is to merely represent something, and not to look like that something, then there is absolutely nothing questionable about the act of symbolization. (Cryptography must be a very questionable practice.)

If there’s an infinite line of green apples in front of you and you say “Look, there’s an infinite line of green apples in front me!” the statement isn’t false by virtue of not looking like what it represents. The word “true” does not mean “a symbol that looks exactly like that which it is trying to represent”. It merely means “a symbol that can be used to represent that which it is trying to represent”.

You didn’t explain why.

Here’s the problem. You’re saying that one can only pretend that the symbol (\infty) can be used to represent infinity. But that’s not true. I don’t have to pretend. The symbol (\infty) CAN be used to represent infinity without any sort of pretense.

So I was right when I said that you’re one of those people who think that the symbol must look like the symbolized in order to be able to say that the symbol represents the symbolized. According to you, if the symbol does not look like the symbolized, you can’t say the symbol represents the symbolized, but you can pretend that it does. Useful contradictions and all. Beside being wrong, what you’re doing here is justifying contradictions in the name of utility.

The symbol (\infty) does not represent undefinability. It represents infinity. Infinity and undefinability are two different concepts.

When I use a symbol to represent something (e.g. an infinite line of green applies), I am not describing what that symbol means, I am simply using a symbol to represent that something.

There is absoultely nothing that cannot be represented using a symbol. All it takes is to pick a symbol (you can literally pick anything) and say “This symbol represents this thing”.

The problem is that you do not understand what it means to say that a symbol is representing something. You have this naive idea that to represent something means to find a symbol that looks exactly like that something. That’s not true.

In other words, you think you’re an expert semiotician while in reality you know nothing about semiotics.

They are different concepts.

That merely means you’re one of those people who do not think but merely follow whatever is popular at the time.

You are deferring to the authority precisely because you have no idea why (0.\dot9 = 1).

I am not using the yellow rectangle to shift the purple one. Rather, the purple one is where it is because of the position of its equivalent rectangle (the red one.)

The purple rectangle represents the red one and the yellow one represents the blue one. The blue one and the red one are independent from each other – they have no terms in common.

You have yet to show me my mistake.

The point is that the red rectangle is not (10) times the green rectangle.

Not quite.

They aren’t and you should go back and re-read what I wrote instead of being stubborn.

The word “infinity” does have a definition.

Or maybe it’s the case that you see contradictions when there are none? Perhaps because you do not understand the meaning of words?

Which is completely irrelevant. Just because the limit of an infinite product is (0) does not mean the infinite product itself is (0).

Not sure what it means to say that an infinite line of green apples is “more boundless”. I certainly didn’t say such a thing. I said that the number of apples is greater than before.

Yes, it’s endless. I certainly did not say it’s finite. My point is that the number of apples is greater.

Correct. But something did happen to the number of apples.

“The quality of having no quantity” is your own idiosyncrasy that has nothing to do with the standard definition of the word “infinite”.

Here we go again. One assertion after another, no arguments whatsoever.

What do you think you can achieve by merely restating your beliefs?

Sure, but how are you going to deduce that one set is bigger than the other?

You added a red apple to an infinite line of green apples. It logically follows that you increased the length of the line.

  1. You have an infinite line of green apples in front of you.
  2. You add one red apple at the beginning of the line. The line is otherwise unchanged.
  3. If the number of apples is the same as before, it follows that you didn’t add the red apple (contradicts premise #2) or that you did add the red apple but that some other apple was removed (also contradicts premise #2.)

Where’s the flaw?

Magnus,

I have a story to tell you. I hate last minute appointment changes more than most people. I’m not a very spontaneous person to that regard. But! If I get early warning, I can psychologically prepare for it and be fine.

The reason I told you this story is because you’re going to lose this debate, like I stated earlier, we’re only getting started. You can thank me for the “heads up” later.

————————

So here’s the deal Magnus, your dot argument is a magic trick.

Here’s why.

One set: 1,2,3,4,5,6,7.8.9…

Another set 1,3,5,7,9,11,13…

One was extracted from the other …

So you say that there’s:

02040608…

And the new set is only

1,3,5,7,9,11,13…

That’s not true.

The other set is:

1,0,3,0,5,0,7,0,9,0…

Etc…

They are still in 1:1 correspondence!

Does the patter indicate a changing relationship as x goes toward infinity?

Given two infinite sets
A = {1,2,3,4,5,6…}
B = {1,3,5,7,9,11…}

When you take the difference;
C = {0,1,2,3,4,5,…}

We can see that the difference between them is increasing as x goes toward infinity. That would indicate that if that pattern doesn’t change, and I don’t see any reason why it would, one set is reaching toward infinity faster than the other. And even though both are infinite, one is “more infinite”, a greater degree of infinite, than the other.

It would merely be a convention for keeping such things straight. I still don’t think it should be said to be “shorter” or “longer”, merely “lessor” or “greater” in degree or magnitude perhaps.

Like I haven’t pointed out the flaw numerous times before.

The flaw is in #3–your assumption that adding an apple to an infinite set of apples makes the set larger. (And BTW, adding the red apple but removing another apple doesn’t contradict #2, it just means something was left out.)

I’ll go deeper:

Let’s say with infinite sets, the size of the set is undefined, then you can’t increase the size by adding one because, well, how do you increase undefined?

If you don’t like that, then let’s say the size of the set is infinite. Then:

(\infty) + 1 = (\infty)

The size doesn’t change.

Want me to go deeper?

You know what? No! No more deeper for you. You need to be convincing me that what applies to finite sets applies to infinite sets. Let it be known that I agree with your logic in regards to finite sets. So let’s not keep beating that dead horse. Focus your energies on proving that what applies to finite sets also applies to infinite sets.

^ See what I did there? Burden of proof thing. That’s right, bitch. :smiley:

Did you really just ignore this whole post ???

Besides, how do you “reach infinity faster”?!

viewtopic.php?p=2755866#p2755866

Gib,

Magnus already addressed that counter argument with:

To say that you “ADD” something is to say that you made it larger, longer, taller, or whatever fits.

What you are saying is that you cannot add to an infinite set.

Couple things:

  1. When we talk about a set being infinite, we mean it has an infinite number of members, not that the value of each member is closer or further from infinity. So we don’t say the size of set {1, 2, 3} is less than the size of set {2, 4, 6}. They’re both 3 members in size.

  2. The rate at which members of a set approach infinity is different from the size of the set. You can say the rate with which set B approaches infinity is greater than the rate with which set A approaches infinity, but that doesn’t mean set B has more members than A.

My argument addressed this better Than gibs approach, yet you ignore it:

The “highest” order of infinity is:

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

Using this argument, we can determine that if you split it, you’re left with:

1.) rational number
2.) 0
3.) rational number
4.) 0

Etc…

1.) 0
2.) uncounted infinity
3.) 0
4.) uncounted infinity

Etc…

No matter how you divide from the “highest order” the sets will all be in 1:1 correspondence.

Done.

Proof.

QED.

You guys lost.

Now continue your meaningless discussions.

Well okay but just take any one value of the sets as a count for a new set. The new set for B will always be longer than the new set for A. So as x goes to infinity, the new set for B will be infinitely larger in count or longer in “length” than the new set for A, even though they are both infinite.

The pattern of always being longer never changes.

You’re still ignoring me.

Do you want the answer to this or not ?

viewtopic.php?p=2755884&sid=dfdee997b272ae54ab79e6c6362c2cb0#p2755884

I ignore it because I can’t make any sense out of it.

The sense of it is that you can do concept replacements of numbers, and order the concepts in a list.

I’m not listing an actual number here, I’m using that abstract representation of the number with words, to place 1:1 correspondence for the “highest order of infinity”

I call it my cheat.

If mathematicians balk at my abstractions, they can only look at how others balk at theirs.

This is fair game what my proof is.

You haven’t.

If you stop pretending that you did, you will stop being frustrated (which means you will stop posting funny gifs that do nothing but reduce the quality of the discussion.)

I know that frustration is a fashionable thing but that does not mean it’s a positive thing.

It does because premise #2 also states that the line didn’t change in any other way (which means that no green apple was removed from it.)

First of all, the size of infinite sets is not undefined.

To say that a size is undefined is to say that it’s unspecified i.e. that it’s unknown.

But to say that a set is infinite is to say that the number of its elements is greater than every finite number. By definition, every infinite set is larger than every finite set. So it’s not like we know absolutely nothing about the size of infinite sets.

But even if the number of things is undefined (i.e. unknown), by adding one to it you necessarily increase it.

Any number + 1 < Result

That’s what the word “add” means.

That’s not an argument.

Magnus can define things however he wants. Doesn’t make it coherent.

Look, I can define subtraction as “making a ham and cheese sandwich”. So then 4 - 2 = … what? What does that mean now that I’ve defined subtraction as making a ham and cheese sandwich? It’s incoherent, right? Just the same, I find the notion of making an infinite set larger, longer, etc. incoherent. I’ve been trying to help Magnus anchor his points by asking him to define what it means for an infinite set to be “bigger” (or whatever) than another infinite set, but the last response he gave me was: an infinite set is bigger than another if the number of units that constitutes it is more. Useful definition for finite sets but just as problematic for infinite sets.

There’s a common misconception of what “addition” means in mathematics. You see it when people argue that 1 + 1 sometimes equals 1, as in when two raindrops join to make one large raindrop. My pet peeve with this is they’re defining “addition” as a process or something going through a change. But when mathematicians talk about addition, they mean something more static, like if you took a photo of the two raindrops, how many raindrops would there be all together in the photo. Addition in mathematics is a way of talking about equivalent meanings, not the result of a process. 2 + 2 means 4. That is, to say there are two objects and there are two other objects is just to say there are four objects. We’re not talking about what happens to those objects, whether they “come togther”, or they meld to become one giant object, or when they come together they collide and break into a million pieces, etc. It’s about how quantities can be expressed in different combinations and permutations.

Magnus’s definition makes sense in terms of a process. If you add an apple to a line of apples, the line becomes longer (at least in the finite case). But that’s “add” as in “you take the apple and place it at the front of the line such that it becomes part of the line.” To that I say, of course you can add to an infinite set. You can drop an apple into a barrel of infinity apples. But as a mathematical concept, it’s incoherent. If you “add” one apple to an infinity of apples, in the mathematical sense, you just get infinity again. What it means for that resultant infinity to be “larger” than the original infinity is what I’m grappling with and what Magnus hasn’t been able to help me with. So far, it remains incoherent.

I did like your definition of infinity as a property of a set, rather than a quantity. I think that’s right. It’s the property of being endless. But as such, it’s pretty binary. A set is either endless or it’s not. I don’t know what “less” endless means.

Not sure whay you mean? You mean something like this:

A = {1, 2, 3,…} ==> A’ = {{x}, {xx}, {xxx}…}
B = {2, 4, 6,…} ==> B’ = {{xx}, {xxxx}, {xxxxxx}…}

They’re both still infinite, and the size of each subset doesn’t determine the size of the parent set.

The size of {{x}, {x}, {x}} is the same as {{xx}, {xx}, {xx}}. They’re both 3.

On the other hand, if you want to do this:

A = {1, 2, 3,…} ==> A’ = {x, … x, x, … x, x, x,…} = {x, x, x, x, x, x,…}
B = {2, 4, 6,…} ==> B’ = {x,x, … x, x, x, x, … x, x, x, x, x, x…} = {x,x, x, x, x, x, x, x, x, x, x, x…}

Both sets are still infinite, but set B appears to be growing faster than set A. This assumes that the time it takes to replace each number in each set with a bunch of x’s is the same, but yeah, the rates would be different. But that doesn’t tell you anything about the size of each set.

Magnus argues that infinity + 1 equals a greater value.

Where did the 1 come from?

An infinite set:

0(because the 1 is gone now) 2,3,4,5,6,7,8,9…

Equals

1,0,0,0,0,0,0,0,0…

In 1:1 correspondence

Otherwise the 1 would have to be tacked in the end of an infinite sequence and would never be expressed :

0,1,2,3,4,5,6,7… 1!

That’s absurd!

I’m actually not that frustrated. The giffies are for comic relief.

Or that it doesn’t make sense.

I’ll concede to this. But I still don’t know how it makes sense to say an infinite set is larger than another infinite set. The whole reason why every infinite set is larger than any finite set is that you can’t get any larger than infinity.

Oh, and just reasserting “…by adding one to it you necessarily increase it” over and over is?

Asserting (\infty) + 1 = (\infty) is just the answer to your question. It’s the reason why adding an apple to the infinite line of apples doesn’t make it bigger. I could go deeper, but I have a feel no depth would ever convince you (you could look up my replies to obsrvr for a hint).

And besides, I’m not doing that. You convince me that what applies to finite sets applies to infinite sets.