Is 1 = 0.999... ? Really?

(0.\dot01) or (0.000\dotso1) is merely a convenient representation of the infinite product ({\displaystyle \prod_{n=1}^{\infty} \frac{1}{10}} = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots ). The infinite product has no last term. The end is merely in the symbol representing the infinite product.

Let’s say you’re standing in front of an infinite line of green apples and one day you decide to add a red apple to it. Where did you add it? We can’t say exactly where because we don’t have enough information but we can say that it’s somewhere inside the line. It can be literally anywhere in the line. It can be at the beginning of the line or it can be 100 apples away from the beginning. One thing is sure: you didn’t put it next to the last apple in the line because there is no such thing as “the last apple in the line”. The point is that, just because you added a thing to an infinite line of things, it does not mean you placed it right after the last thing in the line.

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

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

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

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

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

What does it mean that sets have “finite constraints”?

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.

Omfg :laughing: I know why you put it there!! #-o 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.

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.

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.

Right. (A = {1, 2, 3, \dotso}\neq{B} = {1, 2, 3, \dotso})

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.

No it really very super isn’t! :laughing:

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?

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”?

Yes, it is utter nonsense to say the word apple doesn’t represent apples because it looks nothing like apples.
No I’m not “one of those people who think that” - that’s retarded. Who are these people you’ve been hanging around? I think they’ve had a bad effect on you, or at least your “reasoning” is explained by the company you’ve been keeping.

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…

I’m saying the purpose of matching things that don’t otherwise match is for utility. As you say, it’s not true that the signifier has to match the signified. Hence the distinction between truth and utility.
I hesitate to introduce you to Experientialism, my own original philosophy, which notes this distinction as one of its most primary of tenets. I think it’ll just confuse you even more than you already are.

Exactly! Just as I mentioned last week.

Hey, maybe you have been listening…
I’m sure it’s just an accident, but keep it up even if it is.

The number of finite constraints around natural numbers differentiate them from integers even though there is only one way that each set can be infinite! Exactly.
Hence the negative integers bolted onto the zero that precedes the natural numbers doesn’t “double the size of the infinity”, it subtracts 1 finite constraint.

I have a dream… that we’re getting somewhere! Pinch me, Magnus.

Just so you know, you don’t need to state the n value when it has no bearing or mention on the infinite series.

A better notation might be (\prod_{n=1}^\infty\frac1{10_n}) or maybe even (\prod_{n=1}^\infty{10_n}^{-1}). N could even equal 0, it makes no difference.

Either way, (\lim_{n\to\infty}) of this infinite product is (0).
There is no other value that it approaches, because there’s always a next term that makes it 10 times less, and there is no “smallest” quantity at the “end” of an infinite series.
We can pretend there is, with Epsilon, ε, but we’re only pretending.

One apple at the “starting bound” of a “boundless” line of green apples doesn’t make it “more boundless”.
You could pretend the whole line moved towards you, or u took an apple-length step towards it to produce the same superficial appearance. It’s endless either way - superficials don’t change this. You added a quantity of 1 apple and nothing changed to the quality of endlessness - I’ve been saying this from the very beginning. The quality of having no quantity is not a quantity. 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.

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.

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.

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.

What do you get when you take (1 + 1 + 1 + \cdots) and add one more term to it? You get ((1 + 1 + 1 + \cdots) + \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 + \cdots) 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).

What, you mean you just start enumerating the contents? I suppose you’re saying you enumerate the contents in both sets, and for every member you enumerate in the one set, you map it to a member in the other set, and that mapping counts as the enumerating of members in the other set. So you start with a1, and you map that to a member in the other set, and you call it b1. Then you do the same for a2, labeling it’s mapped counterpart b2. And so on.

In other words, there’s no rule per se–no order with which you must enumerate and map the members–the order/rule is derived based on the enumeration and mapping. So the rule becomes: a1 maps to b1, a2 to b2, etc. The rule is essentially: don’t mess up that order. Is this in the ball park?

So the mapping is based on identity. You’re saying you can’t just take b2 from line B and relabel it b1. You’re saying b2 is a specific point, not the same point b1 was (even though it may now occupy the position b1 once occupied), and therefore cannot be said to be identical to b1. It follows, therefore that the entire line cannot be said to be identical to what it was before (even though, by all accounts, it looks the same).

You must concede then that we cannot say line A is identical to line B even initially, for a1 is not the same point as b1, and therefore even though they look identical, they are not.

This misses the point, of course. The point of this thought exercise was to show that if we’re saying line B is the same as line A based on structure, then it must remain the same after removing points and shifting the remaining points because the structure of line B remains the same. ← This argument called them identical, not because every point in the one line was the same point in the other line, but because their structure is the same. It’s like if I said two buildings are the same because they have the exact same structure and appearance. I don’t mean to say every brick in the one build is a brick in the other building–that would make them a single building–I mean to say there is no way to distinguish them. You could even remove every odd brick in the one building and replace them with entirely new bricks. As long as the structure and appearance remained the same, we can still say they are identical.

Mapping has nothing to do with it. I could map brick a1 in building A to brick b1 in building B, and if b1 was one of the bricks removed, the mapping is broken. Whatever brick we bring in to replace it is not b1, and therefore requires a different label (say c1), and if we want to map a1 to c1, we’d have to concede that it is a different mapping from that between a1 and b1. But as you can see, this is completely irrelevant. We can still say the buildings are identical.

But you’re argument does revolve around identity, doesn’t it?

Then there must be a misunderstanding about what it means to say “the way elements appear in one set is the way they appear in other set”. I took it to mean, the way they appear visually. So when I visualize lines A and B, they appear the same. Then when I visualize them again after removing the odd points from line B, the obviously don’t appear the same. Finally, when I visualize them after shifting the points in line B to fill the gaps, they appear the same again. Or do you mean how they appear with the labels? Labeling a sequence of points P1, P2, P3, P4, P5 would certainly make them look different from an identical sequence of points labeled P2, P4, P6, P8, P10. But labels are accessories. It would be like saying building A and building B are no longer identical when we label them “building A” and “building B”. ← This misses the point of what we mean when we say two things are identical… unless, of course, you’re talking about identity. I’ll have to see from your response.

You did agree above that the labeling is arbitrary. You must have meant only initially (i.e. when we first enumerate the members of each set and determine their mapping). If you mean to say the mapping must remain consistent after that, I see your point.

That is the logical step that is mistaken. The problem here isn’t that I’m failing to point out the flaw, it’s that you don’t understand why it’s a flaw. And I’ve been trying, Magnus, believe me, but it’s not easy. There’s a huge difference between that and failing to even attempt to point out your flaws.

There’s only two options here. Either they’re equal in (infinite) size or the notion of size is meaningless with respect to infinite things (this is the “undefined” size view). In either case, they definitely appear just as long (which was what you were questioning).

a level of stupid i have not seen in a long time.jpg

H’oh, boy. Let’s see if I can use an analogy to explain what’s so deeply wrong with that statement. Two analogies. First, (0.\dot9) is like (0.\dot3) in that they both have infinite decimal expansions. But (0.\dot3) is not an infinite quantity. It’s just one third. (I chose (0.\dot3) because I can’t say (0.\dot9) is just 1 without you fighting me on it.) Mistaking the quantity represented with the representation is such a juvenile mistake. But just in case you mean something a little more sophisticated, I have a second analogy for you. By “infinite sum” I’m guessing you mean 0.9 + 0.09 + 0.009… In that case, both terms are the results of infinite sums. But then adding infinite sums is not the same as adding infinite quantities. So here’s the second analogy: 10 + 20 + 30 is a finite sum. It equals 60. What makes it finite is the number of terms. There are 3 of them. But you’re not adding 3. You don’t include the number of terms as one of the terms being added. You don’t add 10 + 20 + 30, then say: well, that’s 3 terms, gotta add 3 for 63. So if we had an infinite sum, like 0.9 + 0.09 + 0.009…, why would we add infinity? The sum of the terms is just 0.999… which is just 1, a finite number (or even if you disagree with that, it’s a number less than 1, but still definitely finite). With (9.\dot9 - 0.\dot9 = 9), you’re not subtracting infinity from infinity. You’re subtracting two very small finite numbers. You may have had to sum an infinite number of terms to get each one, but even that isn’t adding infinities. But regardless of whether infinity is a quantity or not, I’ve never come across a case of doing arithmetic with infinity and getting meaningful results.

You can say they’re both undefined, or both infinite. ← Those are the only two senses in which they are equal. You certainly can’t say they are unequal. On what basis could you say they are unequal? And which one is less/more than the other, and by how much?

Anywaaay, I want to get back to your mapping argument. Though I see a few flaws in it (not the least of which is that it misses the point), I think I understand what you’re trying to get at. You’re trying to get me to keep track of which point in the one line corresponds to which point in the other. You’re hoping to show me that if points that were side by side initially are no longer side by side in the end, then I might realize: wow, there is a difference after all. But even if I grant that, that makes no difference to the size of the lines (which was my point). Even if it’s no longer the same points side by side, the lines still appear the same, and appearances is all I need. To judge the length of something as equal to something else is all a matter of appearance. We say: the brick appears to be 10 inches long because it appears to be the same length as the ruler put next to it, at least up to the 10-inch mark. In fact, my argument is bolstered by the fact that the points end up being different. The point was that you can shift a whole ream of points down the line and it will never change the length of the line.

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.

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? :smiley: lol

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.

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.

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.

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?

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

See above ^

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.

You think that Bertrand Russell was missing something upstairs…

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

Oh wow…

You two guys…

Sure I do, mate :slight_smile: 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.

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

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

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!

:laughing:

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

Thanks for your reliable non-contributions, MagsJ.

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

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

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.

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:

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

You’re asking for more proof than necessary.

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. ; )

Oh I contribute plenty! but you’ll never know… coz I never tell. :-$

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

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?

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

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

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”

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…

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.

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.

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.

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.

Yeah.
It’s not necessary to prove the crux of your argument.
You’re right… :icon-rolleyes:

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.

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

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.