Is 1 = 0.999... ? Really?

[Silhouette]

I criticised the use of "infinite" in reference to sets when any aspect of their construction is a product of finitude.

What does it mean to say that an aspect of a construction of some set is a product of finitude?

Take a standard infinite sum \(\sum_{n=0}^\infty{a_n}\)
The start of the sum is defined as 0.
The function "a" is defined and its relation to the starting point is defined by its subscript of n.
The big Sigma defines what we're doing to each term in the series in relation to the others.
All these specified finite constraints are set up using finite notations of finite quantities.
They all come together in one standard construction.

And there's 1 notation that is of an infinite.
All these finite aspects to this construction, and there is 1 aspect to this construction that is infinite.
The finite aspects of this construction are a product of finitude. The infinite aspect is a product of infinitude.

That's what I mean.

I mentioned how even the natural numbers have a finite starting point on the number line

The set of natural numbers does not have a starting point. There is no first element, member, number.

Wow.

Again you distract away from the "well-ordering" of the number line that I was talking about to this less precise "set of natural numbers" with cardinality only.

So dishonest/ignorant. You pick.

never mind the line being finitely bounded in all other dimensions as well

How is the set of natural numbers (not the number line) bounded in all other dimensions?

This is obvious and something I've explained many times.
No doubt you're denying its existence by denying the well-orderedness of natural numbers.

Again you're distracting from my point about the number line just to evade addressing yet another one of the many flaws in your line of reasoning.

being infinite in only one dimension in one direction

How can sets be infinite in more than one dimension and in more than one direction?

Easily.

See how the well-ordered number line of natural numbers doesn't go "backwards" from 1, and it only goes "forwards"? And it only goes forwards along 1 dimension? Perhaps horizontal, like the x axis, or maybe even vertical, like the y axis if that's more convenient: the line goes along only one dimension either way. This is one direction along one dimension only, and starting from the finite quantity of 1. That 1 is a finite constraint on the start. The direction of the progression of the number line is a finite constraint on it progressing along any other dimension than the line it follows. The only thing that's infinite about it is the fact that once it starts, it keeps going in that one direction along that one dimension indefinitely.

How can you not understand this?

It's finite in many more ways than it is infinite, yet it's still called infinite because it's infinite in at least one way.

There is only one way that sets (including the set of natural numbers) can be finite or infinite.

Yes. That's what I've been saying this whole time.

In other words, any size of "infinite sets" is determined by their relative lack of finite constraints and not any "different size of infinity".

What does it mean that sets have "finite constraints"?

See above ^

It's only if you could remove all finite constraints to "infinite" sets, that you'd get an entirely infinite set, which would mean "boundless in every way one can think of".

A set is said to be entirely infinite if the number of its elements is endless. That's what it means for a set to be entirely infinite. A set cannot be more or less infinite. It cannot be partially infinite. It's either infinite or it is not.

A set cannot be more or less infinite!!!!!!!!!!!!!!!!!!

You just said it!!!! THANK YOU.

Thank god this fucking joke of a topic is over on "sizes" of infinity.

Principia Mathematica by Alfred North Whitehead and Bertrand Russell.

Building up the foundations of maths, they finally reach 1+1=2 by page 362 (in the 2nd addition at least, in the 1st addition it's page 379).

So you are telling us that someone took 362 pages just to prove that 1+1=2?

First, I don't believe it. And then if they did they were definitely missing something upstairs.

You think that Bertrand Russell was missing something upstairs...

Talk about the fallacy of personal incredulity, guys!!!!!!

Oh wow...

You two guys...

Goddamnit, why is it so impossible for amateurs to believe that someone else might know things they don't?
Dunning Kruger effect, obviously. But fuck me is it frustrating to have to deal with.

I realize that appearances aren't everything and no offense but compared to Magnus, you are the one who appears to be the amateur here suffering from Dunning-Kruger effect.

As he pointed out earlier, you have not shown any flaw in his argument. You just say he is wrong and then give your own narrative. If Whitehead and Russel argue like that, I can see why it took so long for them to do so little.

Sure I do, mate Our previous encounters on infinity make so much more sense now you've revealed your ignorance about Russell and mathematics.
What are you doing on a philosophy board again?
Fair play if you actually want to learn from your lesser positioning, but all this time you've actually been professing expertise and credibility just like Magnus - the other charlatan of the same kind. Oh internet..., how do you manage to bring out the quacks so efficiently and effectively?

I have thoroughly shown very many flaws in all arguments against \(1=0.\dot(9)\) and all the other peripheral arguments brought up around that position.

Now that you've revealed you have negligible background or capability on this kind of subject, my suspicions have been confirmed that it is likely sufficient to regard your opinions on my reasoning about it as null and void.

Here it is:

\(\frac{9 + 0.\dot9}{10}=0.\dot9 + \underline{\text{the missing term}}\)

If you're asking me about its value, I don't know, I didn't calculate it. Do you really think such is necessary in order to prove that there is indeed a missing term? I don't think so.

Well how about you fucking calculate it, eh? Magnus?

Come back to me when you actually have something, huh?

What do you get when you take \(1 + 1 + 1 + \dotso\) and add one more term to it? You get \((1 + 1 + 1 + \dotso) + \underline{1}\). The underlined is the added term. That's where it is. In this particular case, it's pretty easy to calculate the value of the added term because every term in the infinite sum \(1 + 1 + 1 + \dotso\) is equal to every other. This isn't the case with \(0.\dot9\), so figuring out the value of a single term is not so straightforward. You'd have to find an equivalent infinite sum where every term is equal to every other and then calculate how many terms of that sum is equal to a single term of \(0.\dot9\)

So add 1 to an endless string of 1s. Is that all you have?
Is it "more endless" than "endless" now?
Is the "size" of the number line of natural numbers bigger now?

So basically the extra 1 is nowhere, right? It's nowhere, thats where it is.
Add 1 to any finite set, and sure - you're absolutely perfectly unequivocally correct!

Don't you know about second infinity, Silhouette? Ask Magnus all about it.

Still following..

I will adject when I see fit, but right now my money is still on Magnus, as man’s held his ground and not swayed from his thinking. No pressure though bro, ok? lol

Yay, the opinion of someone who puts money on stubbornness and neither learning nor adapting anything.

Thanks for your reliable non-contributions, MagsJ.

[Magnus Anderson]

I'm saying the purpose of matching things that don't otherwise match is for utility.

That's true but it's trivial and irrelevant.

I'm saying the whole point of a word is that it isn't what it represents - hence the whole distinction between signifier and signified that I brought up...

That's also true and it's also trivial and it's also irrelevant.

All words obviously have definitions, but the things they refer to can defy definition - making the act definition questionable in the first place.

Here you say that there are things that defy definitions which means that there are things that cannot be defined.

But it's not things that we want to symbolize that we define but our symbols. To define some symbol S is to verbally (or non-verbally) describe its meaning.

If you want to say that there are things that cannot be represented by symbols, that's a different story, but you're still wrong. Anything can be represented using any kind of symbol. All it takes is to pick a symbol and say "This symbol represents that thing".

If all you want to say is that the symbol and the symbolized are two different things, fine, but 1) you're using way too many words, and 2) I don't understand the relevance of that.

You're also claiming that the act of defining words is questionable.

The signified has infinitude, but the signifier implies at least some finitude for it to be a word at all. There's obvious problems with the truth in doing this, but undeniable utility in doing so - hence why people perform this questionable act in the first place.

We're supposed to believe that there is something questionable about the act of using a finite symbol (such as a word) to represent something that is infinite.

And here's more of your obscurantism:

Only in this way with these concessions can the word infinity have definition and even synonyms like boundless, which is only a "definition" in the same way that a tautology gives extra information (it doesn't). There's only an appearance of definition here (which again is your whole problem), and on top of that infinitude is an absence of finitude (definability) rather than a definable thing itself.

You're speaking of concessions, definitions that are merely apparent, the idea that finitude is synonymous with definability and so on.

[Magnus Anderson]

Well how about you fucking calculate it, eh? Magnus?

Come back to me when you actually have something, huh?

You're asking for more proof than necessary.

[MagsJ]

Still following..

I will adject when I see fit, but right now my money is still on Magnus, as man’s held his ground and not swayed from his thinking. No pressure though bro, ok? lol

Yay, the opinion of someone who puts money on stubbornness and neither learning nor adapting anything.

Then be more succinct, and then I/we may take more note, but until then..?

May I ask.. why do you always feel cornered/hounded? I’m very observant you know. ; )

Thanks for your reliable non-contributions, MagsJ.

Oh I contribute plenty! but you’ll never know.. coz I never tell.

I’m tired, so will re-engage on this later.

[obsrvr524]

I have a feeling you think the mapping argument is crucial because, again, that's how it works with finite sets. With finite sets, if you want to know whether two sets contain the same number of members, you enumerate them. You say: that's 1, 2, 3, 4, 5 things in the first set. That's 1, 2, 3, 4, 5 things in the second set. Yep, for every item I counted in the first set, there is an item that I counted in the second set. When I stopped counting the first set, I stopped counting the second--not an item before, not an item after. Enumeration and mapping only work because the sets are finite. They work because there is an end to the enumeration and the mapping that you will eventually attain, and when you do, you will know whether the one set has more members than the other, has less members, or is the same. If you find there are members left over in one set that can't be mapped to members of the other set, that's how you know the first set has more members. All this hinges on an end to the enumeration and the mapping. But if you enumerate an infinite set and you map each member to members of another set as you enumerate that set, you'll never be able to say: well, it appears I'm done counting this set before the other. It appears there's members left over in the other set that can't be mapped. It doesn't matter if you map only every second member, or every third, or every fourth, the result will be the same: it'll go on forever, and skipping every second, third, fourth will never show that there's fewer members in the latter set than in the former.

^ Now don't say I avoid explaining why this leap from finite sets to infinite sets is a flaw.

Isn't it simple logic to deduce that you have a pattern that isn't going to change as you proceed toward infinity? Doesn't calculus do that at its core?

Every item need not be individually counted just as every item need not be printed out. A series is deducible. And a process is deducible, else the sum of convergent series could never be known.



Now that was an example of pointing out a flaw in the other person's argument. When is someone going to point out a flaw in Magnus' argument?

[Magnus Anderson]

Again you distract away from the "well-ordering" of the number line that I was talking about to this less precise "set of natural numbers" with cardinality only.

That's precisely the point: to distract you from your own distractions.

[Magnus Anderson]

Fair play if you actually want to learn from your lesser positioning, but all this time you've actually been professing expertise and credibility just like Magnus - the other charlatan of the same kind. Oh internet..., how do you manage to bring out the quacks so efficiently and effectively?

Noone here is as preoccupied with their self-image as much as you are.

[Ecmandu]

As for calculus, like I stated earlier, it’s a ROUNDING discipline.

When people enter the shitter is to declare an equality for the sequence and call it a “bound infinity”

[gib]

I will adject when I see fit, but right now my money is still on Magnus, as man’s held his ground and not swayed from his thinking.

I agree Magnus wins the stamina race.

[Ecmandu]

I will adject when I see fit, but right now my money is still on Magnus, as man’s held his ground and not swayed from his thinking.

I agree Magnus wins the stamina race.

He also wins the ignoring people race.

I ask him point black, what’s 1/2 infinity... ? he cowers.

The reason he cowers is because of two reasons:

1/2 infinity means nothing. The second reason is because it makes it nonsense to talk about two infinities, they are merely (2) 1/2 infinities, which still only equals a whole infinity.

Magnus needs to be able to say that 2 infinities exist to keep going with the thread...

[Silhouette]

I'm saying the purpose of matching things that don't otherwise match is for utility.

That's true but it's trivial and irrelevant.

I'm saying the whole point of a word is that it isn't what it represents - hence the whole distinction between signifier and signified that I brought up...

That's also true and it's also trivial and it's also irrelevant.

It's absolutely relevant since the core of your argument is to regard the infinite as finite.

I say there is a problem with this.
You agree that words aren't what they represent.
The definite symbol \(\infty\) is not definable - it represents the exact opposite of definable even though the word/symbol/signifier takes the same form as things that are definable.
See, you agree with me and understand the undeniable logic that I'm forwarding but you won't lend this your slightest acknowledgement. So dishonest.

All words obviously have definitions, but the things they refer to can defy definition - making the act definition questionable in the first place.

Here you say that there are things that defy definitions which means that there are things that cannot be defined.

But it's not things that we want to symbolize that we define but our symbols. To define some symbol S is to verbally (or non-verbally) describe its meaning.

If you want to say that there are things that cannot be represented by symbols, that's a different story, but you're still wrong. Anything can be represented using any kind of symbol. All it takes is to pick a symbol and say "This symbol represents that thing".

If all you want to say is that the symbol and the symbolized are two different things, fine, but 1) you're using way too many words, and 2) I don't understand the relevance of that.

You're also claiming that the act of defining words is questionable.

The act of definining words is questionable by virtue of words not equalling what they represent, which is the source of many philosophical misunderstandings and is solved by Experientialism - but this more general concept is beyond the scope of this thread, although the specific instance of a finite symbol representing an infinite quantity is absolutely central to the thread and your misunderstanding of "infinity" as finite.

I'm not saying you can't pretend that the symbol \(\infty\) can be used to signify infinity - clearly it routinely is - but the superficial appearance of a signifier must not be confused with the nature of what it denotes when the nature of the signified is that which defies symbolic representation.

My whole point this whole time is that while you can write \(\infty+1\) etc. and everyone knows that you think you mean by this, the inherent problem of defining the symbol of undefinability completely nullifies the sense we can make of \(\infty+1\).

You're not describing the meaning of infinity by symbolising it as \(\infty+1\), you're inviting the possibility of a whirlwind of misunderstanding. "I picked a symbol to represent that which can't be summed up in a symbol" does not make for a sound foundation. Admit this, and the impending chaos that it inflicts on your points.

You're speaking of concessions, definitions that are merely apparent, the idea that finitude is synonymous with definability and so on.

Let me know in all honesty if you think finitude is at odds with definability.

This will give everyone a very clear indication of how competent you are with semantics: a context with which we can frame this entire debacle such that we may mentally resolve it with immediate finitude.

Noone here is as preoccupied with their self-image as much as you are.

Funny, because I already denied any involvement in the already-established and legitimately accepted proof that \(1=0.\dot9\) and encouraged you all to forget I had any part to play in the fact that it's already been proven by professional mathematicians. I don't want any credit nor hope for any personal gain from the success of people learning why something basic is true.

I give no shits about what you or anyone thinks of me, I post here solely as an exercise to familiarise myself with the kinds of bullshit that irrational and amateur people come up with against good ideas - and even hold open the possibility that they might have thought of something that I haven't in coming up with my own ideas. This is a testing space for me, I don't want friends or respect, but I do respect people who have the intellectual fortitude to recognise closure when it confronts them. I naively hope that this will happen, and sometimes it does, and even though I expect average people to be intellectually deficient I so far have not mastered the ability to refrain from frustration when they inflict their weaknesses upon me. Maybe I'll overcome this weakness of mine one day, but I'm still not fully sure that I need to.

You're asking for more proof than necessary.

Yeah.
It's not necessary to prove the crux of your argument.
You're right...

Then be more succinct, and then I/we may take more note, but until then..?

May I ask.. why do you always feel cornered/hounded? I’m very observant you know. ; )

Succinct replies leave too much to the imagination and satisfy the impatient pleasure principle more than intellectual rigor - I require that subjects are sufficiently and exhaustively dealt with. Brevity is a secondary aim, but it is limited by the degree to which I demand thorough dealings.

I rarely feel cornered, and have not been so far during this thread so your observation skills have unfortunately been mistaken.
If you felt as though I was, then regretably you have misjudged my comprehensiveness as corneredness though you have correctly identified my frustration with my feeling of houndedness by the tedious repetition of falsity.

Thanks for your reliable non-contributions, MagsJ.

Oh I contribute plenty! but you’ll never know.. coz I never tell.

I’m tired, so will re-engage on this later.

Yeah yeah, jam tomorrow.
Funny how your excuses are never followed up - I'll never know the plenty you contribute indeed!

[obsrvr524]

The definite symbol \(\infty\) is not definable - it represents the exact opposite of definable even though the word/symbol/signifier takes the same form as things that are definable.

I don't see how that makes any sense at all. It's very obvious that the symbol \(\infty\) certainly is defined and well known (for centuries).

Additionally infinity itself is defined and well known. The only problem that I see is that it is not defined sufficiently for maths operations to be sensibly used on it.

infinity (ɪnˈfɪnɪtɪ)
n, pl -ties
1. the state or quality of being infinite
2. endless time, space, or quantity
3. an infinitely or indefinitely great number or amount
4. (General Physics) optics photog a point that is far enough away from a lens, mirror, etc, for the light emitted by it to fall in parallel rays on the surface of the lens, etc
5. (General Physics) physics a dimension or quantity of sufficient size to be unaffected by finite variations
6. (Mathematics) maths the concept of a value greater than any finite numerical value
7. (Mathematics) a distant ideal point at which two parallel lines are assumed to meet
Symbol (for senses 4–7): ∞

[Magnus Anderson]

It's absolutely relevant since the core of your argument is to regard the infinite as finite.

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

The definite symbol \(\infty\) is not definable - it represents the exact opposite of definable even though the word/symbol/signifier takes the same form as things that are definable.

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

The act of definining words is questionable by virtue of words not equalling what they represent

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".

the specific instance of a finite symbol representing an infinite quantity is absolutely central to the thread and your misunderstanding of "infinity" as finite

You didn't explain why.

I'm not saying you can't pretend that the symbol \(\infty\) can be used to signify infinity - clearly it routinely is - but the superficial appearance of a signifier must not be confused with the nature of what it denotes when the nature of the signified is that which defies symbolic representation.

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.

My whole point this whole time is that while you can write \(\infty+1\) etc. and everyone knows that you think you mean by this, the inherent problem of defining the symbol of undefinability completely nullifies the sense we can make of \(\infty+1\).

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

You're not describing the meaning of infinity by symbolising it as \(\infty+1\), you're inviting the possibility of a whirlwind of misunderstanding.

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.

"I picked a symbol to represent that which can't be summed up in a symbol" does not make for a sound foundation.

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.

Let me know in all honesty if you think finitude is at odds with definability.

They are different concepts.

I don't want any credit nor hope for any personal gain from the success of people learning why something basic is true.

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 \(0.\dot9 = 1\).

[Magnus Anderson]

It's just idiotic to use that as an excuse to shift over the separate purple set that doesn't even include the yellow set that's being used to superficially shift it.

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.

This shit you've just wasted keyboard strokes on is not only unnecessary waste, it does nothing to excuse the fact that you've started a set "one to the right" just to say that it doesn't start "one to the left" like the green one.

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

I know WHY you did it, it's painfully obvious, but the result of playing around with superificial positionings doesn't change the fact that green and purple have perfect bijection: the first terms match, the second do too etc. It's just more sophistry by the uninitiated to fool the uninitiated.

Not quite.

What am I not listening to when you literally write out the exact same set twice? You saying they're not the same even though they are?
So sorry for not believing you when you literally write out right in front of everyone the exact same set.

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

How can you not see that the definition of definition can't apply to that which has no definition? And yet we do it anyway?

The word "infinity" does have a definition.

Presumably you think it's nonsense because you lack the ability to see contradictions plain and simple right in front of you, even in your own "reasoning"?

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

Either way, \(\lim_{n\to\infty}\) of this infinite product is \(0\).

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

One apple at the "starting bound" of a "boundless" line of green apples doesn't make it "more boundless".

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.

It's endless either way - superficials don't change this.

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

You added a quantity of 1 apple and nothing changed to the quality of endlessness - I've been saying this from the very beginning.

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

The quality of having no quantity is not a quantity.

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

Add it literally anywhere in the line, as you say - no difference. It would make a difference to a finite line, for sure. Adding a new first element to an infinite set just gives you an infinite set with a new finite bound - the finite "1st element" changed, shifting all successors down by 1 place infinitely.... - no size change occurs. It would occur for a finite set, sure, but you can't have a "longer" infinite endlessness even if you change a finite constraint to how it starts.

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

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

[gib]

Isn't it simple logic to deduce that you have a pattern that isn't going to change as you proceed toward infinity? Doesn't calculus do that at its core?

Every item need not be individually counted just as every item need not be printed out. A series is deducible. And a process is deducible, else the sum of convergent series could never be known.

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

[Magnus Anderson]

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

[Magnus Anderson]

That is the logical step that is mistaken.

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?

[Ecmandu]

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!

[obsrvr524]

Isn't it simple logic to deduce that you have a pattern that isn't going to change as you proceed toward infinity? Doesn't calculus do that at its core?

Every item need not be individually counted just as every item need not be printed out. A series is deducible. And a process is deducible, else the sum of convergent series could never be known.

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

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.

[gib]

That is the logical step that is mistaken.

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?

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.

[Ecmandu]

Isn't it simple logic to deduce that you have a pattern that isn't going to change as you proceed toward infinity? Doesn't calculus do that at its core?

Every item need not be individually counted just as every item need not be printed out. A series is deducible. And a process is deducible, else the sum of convergent series could never be known.

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

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.

Did you really just ignore this whole post ???

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

viewtopic.php?p=2755866#p2755866

[obsrvr524]

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.

[gib]

Isn't it simple logic to deduce that you have a pattern that isn't going to change as you proceed toward infinity? Doesn't calculus do that at its core?

Every item need not be individually counted just as every item need not be printed out. A series is deducible. And a process is deducible, else the sum of convergent series could never be known.

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

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.

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.

[Ecmandu]

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.

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.

[obsrvr524]

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.

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.