Is 1 = 0.999... ? Really?

I know.

Can’t reach it with an infinite number of steps either. There’s always infinitely more when it comes to infinity…

Believe it or not, infinity literally means no end - clue is in the word.
One might even say this is only the infinity’th time I’ve pointed that out…, which I guess is why we’ll never get there :wink:

If you say “There is an infinite line of apples in front Joe” that does not mean that you cannot add “One day Joe ate them all”. The two statements aren’t contradicting each other. The second statement is not implicitly stating that there was no infinite number of apples in the first place.

If “One day Joe ate them all”, and “There is an infinite line of apples”, there would be always be more left to eat if the line was infinite. A contradiction.

There is no final bound to endlessness that allows the completion of “all”.
The series is only “complete” in that it is consistent and predictable in its infinite progression.

Math works by defining from the finite down toward this infinite amount of divisibles.

If it tried to work starting from an infinite set between two finite integers it would never get off the ground, never mind getting from one finite bound to the other - Zeno style.
This is only expected, due to the contradiction in the premises of entirely bounding a set (with finites) and calling that set boundless (infinite).

You solve Zeno by defining from finites (hence the tautology in terms) instead of defining from infinites (a contradiction in terms).
That way you eventually get to the problems of infinity anyway, but you get there via a structure that you wouldn’t have if you worked from the problems of infinity.

This has been the point I stated right at the start of my contributions to this thread, but due to others getting bogged down in various details that are confusing them, discussion about my whole point never really began.
Arising from a fundamental tenet of my own original philosophy, Experientialism, utility (e.g. finites) defies truth (e.g. infinites) at a basic level - i.e. experience is continuous: there are no gaps of nothingness in it to genuinely separate “discrete experiences” from one another. Yet without treating Continuous Experience as “discrete experiences” we can’t do anything useful with it, or even say anything at all. You need to artificially dissect continuity in order to garner knowledge from it “about it”, even though any subsequent analysis resulting from this inevitably leads to problems such as those presented by (0…1) anyway a posteriori - as is consistent with the self-evident experience as continuous “before” knowledge a priori that you started with in the first place.

I solve Zeno by taking a fucking step forward.
It’s not an infinite space. It’s an infinitely divisible space.
Likewise, I can divide an apple infinitely, and that does not mean I have infinite apples.
I can spend an infinity of time eating it by always taking a bite that is half of what’s in my hand, but I still only ate (slightly less than) half of an apple.

To play with Zeno’s idea in an infinite space, where do you set the middle point?

seems obvious

I see… I’d say, that infinite objects can only exist in an infinite space (universe), but concepts are not bound by these physical limitations… obviously. So adding more apples to a set, could not then make that set become infinite… it would just make that set larger.

The term infinite is therefore self-defining, so either something is infinite or it is not.

You demonstrate grounds to refute Zeno by taking a step forward (fucking or otherwise), you don’t solve it.
You solve it by identifying the flaws in the premises, correcting them and ending up with Experientialism.

Infinitely divisible space… into what? Infinitesimals? And yet there’s always something smaller tending towards no other value than zero…
Any non-zero value infinitely many times simultaneously satisfies never getting beyond infinitesimal space because the infinitesimal is so small, and getting to infinite space because the infinite is so large. It’s undefined, and treating it with any specificity violates its infinitude - so whilst being both infinitesimal and infinite… it’s neither. Being “undefined”, as the infinite is, is clearly problematic, no?
And yet, if we take the limit of infinite division that is “zero”, even infinite zeroes would not seem to suggest any space at all.

This is why knowledge is defeated from its very start when it tries to begin with the infinite - because knowledge, by definition, needs definition: it needs finitude.

There is no middle point in an infinite space - the same reason why there’s no centre of the universe even though everything’s moving away from everything else. You need relativity to arbitrarily choose a point of reference to measure something against finitely, and even then only get an answer specific to that point of reference. The same thing applies to defining 1+1=2, you have to establish an empty set first: pure conceptual boundary. Only then can you define the successors, equalities and inequalities necessary to define 2. The problem with the infinity in between 1 and 2 (same as the problem between 0 and 1) only crops up after that once one at least thinks they’ve gotten off the ground.

Clearly dividing an apple infinitely does not yield infinite apples.
Eating half an apple, then half of that and half of that, which I believe is what you’re saying, leaves partial products of slightly more than nothing at all and the limit of the process approaches zero apple.

Great! We’re on the same page.
And yet, it’s funny to call something obvious when nobody’s ever thought to suggest it: that utility and truth are at odds with one another.
Everyone has always regarded them as complementary - and no wonder: isn’t it odd that “what works” is based on a deficit of truth?

Yes, if you want to make the thought experimental physical, you’d need an infinite space (universe), and yes, concepts are not bound by these physical limitations.

Adding more apples to a finite set would make that set larger, no issue there at all.
The issue is that adding apples to an infinite set of apples doesn’t make “infinity bigger” because expanding the bounds of something to make it bigger only applies to the bounded and not the boundless.

The term infinite defies definition by the definition of “definition” and of “finite”. Either something is infinite or it is not - of course. Infinite is only “definable” insofar as we can easily define finite… and then saying “not that”. This says what you don’t have and not what you do have. The analogy I used is that this “defines” what’s in a hole by defining the boundaries of the hole and what’s outside of it (i.e. it doesn’t define what’s inside the hole at all).

approaches half apple

it’s not an original thought
all of scientific research operates with this in mind
utility is a compromise

Yeah! I get to play devils advocate to Magnus for a moment!

Magnus writes a number like:

2,4,6, 1… (1 repeating)

Now!

1,1,1,1… (repeating) is possibly 8 smaller in VALUE, but not correspondence!

For example:

3.000…

Is smaller in VALUE than PI.

Correspondence however is the same.

(0 \times 0 \times 0 \times \cdots) is an infinite product. It’s a product made out of an infinite number of terms. It has no end.
Agree?

Even though (0 \times 0 \times 0 \times \cdots) is a product made out of an infinite number of terms, its result is equal to a finite number that is (0).
Agree?

You are certainly not telling us that the result of an infinite product can never be calculated because due to the number of terms being endless there is no end to the process of calculation? I hope that’s not what you’re trying to tell us.

How do we know that the result of this infinite product is (0)? We know it’s (0) because we know that all of its partial products are equal to (0). What this means is that we can calculate the result of an infinite product by looking at its partial products.

So yes, I know full well we’re dealing with an infinite product (and not merely its partial products.) But I also know that the way to calculate the result of an infinite product (or at least, to calculate its bounds) is by analyzing its partial products.

The insight is pretty basic: regardless of how many terms there are in a product of the form (\Pi^{n} 0), the result will always be (0). Now matter how big the number of terms is (indeed, even if its bigger than every finite number), the result will be (0).

Something similar applies to (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots). Here, we can observe that every partial product has a different value. It doesn’t matter how large the number of terms is (even if its larger than every finite number), the result is always different from every other partial product. This alone tells us that the result is not equal to (0). But there is one more thing we can observe: no matter how large the number of terms is, the result is never equal to or below (0). And this is what tells us that the result is greater than (0).

You, on the other hand, are telling us that we can get (0) by taking (\frac{1}{10}) and raising it to a sufficiently large number (namely, that of infinity.)

The point is that any product of the form ((\frac{1}{10})^n) where (n) is any number greater than (0) is greater than (0). The only condition for (n) is to be greater than (0). It can be a finite number but it can also be a number larger than every finite number (i.e. an infinite number.) Either way, the result is greater than (0).

If he ate them all, this means he left no apple behind.

You can say that he can’t eat them all because, by description, the line resists being completely destroyed. So he can eat some (a lot of them, perhaps even an infinite number of them) but he can’t eat all of them. But the two statements, “There is an infinite line of apples” and “One day Joe at them all”, claim no such thing. The word “infinite” certainly does not mean “incapable of being completely changed or destroyed”.

Let’s take a simpler example.

“Once upon a time, there was an infinite number of apples at some point in the universe. One day, every single one of them disappeared.”

Are you telling us that the process of an infinite number of apples disappearing cannot be completed within a finite period of time (not to mention instantly)?

How about this example?

“Once upon a time, there was an infinite number of green apples at some point in the universe. One day, every single one of them changed to red.”

Magnus, you need to understand when someone is honestly trying to help you.

As a process 1/10 always leaves a 1 at the end, as an infinity it does not!

I’ve told you twice in this thread already that you treat infinities as objects OR processes as it suits your needs in the current post.

The thing is:

You can’t do this shit, you need to pick a real side and debate it. Thus far, your posts have been non debates.

For the life of me I can’t make sense of this from your wording as approaching half an apple… must be my fault, right? Demonstration please?

(\prod_{k=0}^\infty(\frac1{2})^k=0)
Type “infinite product of (1/2)^k” into Wolfram Alpha.
It’s the same for any fraction less than 1 (starting with 1 whole apple and proceding to eat any possible fraction of it) and for any k value as a starting point… - you always get zero.

Eat half of a whole apple, you have half left
Eat half of the remaining half, you have a quarter left
Eat half of the remaining quarter, you have an eighth left
Etc. down to the limit of nothingness…

The scientific method is indeed a process of relative improvement, rather than a sure way to arrive at absolute truth - but like dialectics, even with the realisation of compromise, it’s intended to tend towards truth or at least a more truthful representation of reality.
Utility being at odds with truth realises that all knowledge, even that refined by the scientific method is in fact a tendency away from the truth of continuity.

I don’t think Magnus meant that by his ellipsis, but I agree with you that 1 repeating would amount to infinity all the same.
If the 2 corresponds with the first 1, the 4 with the second, the 6 with the third, and with all 1s matching after that, the total difference would be 1+3+5=9, not 8. Is that what you meant by the difference in value? Either way any apparent “value difference” from looking at only the start would get swallowed up by the undefined essence of finitude-being-opposed (infinity). The superficial appearance of different finite values at the start does nothing.

I assume by representing (3) as (3.\dot0) and corresponding each decimal place with (\pi), you get bijection and therefore the implication that the quantities are the same size - or at least the cardinality of the sets is the same.
If it can be accepted that adding in the appropriate number of 0s to any set, and that you can therefore theoretically induce bijection for everything and anything, this would throw into question the whole notion of set cardinality being evaluated by bijection.
It’s a big “if” though, but I’m struggling to see any flaws in it other than what I said about quantities that require the quality of infinity to denote them correctly, and those that don’t require it (e.g. recurring 0s either before or after any stated finites).

We know the infinite product of zeroes is zero insofar as we accept that any quantity, whether finite or infinite “zero times” is zero.
The fact that the partial product also comes to zero is a symptom of this, it’s not the reason for any quantity zero times being zero.
Anything zero times being zero is just logic - we don’t need the partial product to know this - that’s just a red herring.

Insofar as we analyse what we get when we get away from partial products to their limit, we are “analysing its partial products” to “calculate the result of an infinite product” - but this “insofar” isn’t very far if it’s exactly what we’re trying to get away from…

Btw your notation (\Pi^{n} 0) made me laugh :laughing:
Try something like (\prod_{n=1}^\infty{0_n}) or probably just (\prod_{n}^\infty{0n})

We know from (\prod_{n=1}^\infty{\frac1{10_n}}) that indeed any partial product has a different value - so that’s no help, until we extrapolate the limit that it tends towards. The partial product also tells us that for any partial product there’s a smaller product - forever. You never get small enough. Anything greater than zero is too large, and the only value that you can’t divide smaller is its limit: zero. So whilst the partial product looks like it’ll never ever get to zero “no matter how large the number of terms is”, zero is still the only value that could make sense - as well as just so happening to be the limit. This is even though (\prod_{n=1}^\infty{0_n}=0) as a separate infinite sum that just so happens to reach the same result - but doesn’t mean the infinite series are the same - only that they “look like” they’re different and “look like” they ought to give different answers. That’s what your point is - that the partial product makes it “look like” it’ll always be greater than zero. But you mistake the same to apply for infinites, which cannot yield an answer small enough unless it reaches its limit of 0. Doesn’t matter how much larger you want this “infinite number” to be than any “finite number”, it will never be small enough - therefore all numbers greater than zero are invalid. Only zero can be valid - I can only apologise on behalf of appearances for fooling you, sorry.

You’re just not appreciating what undefinability does to what’s so clearly definable for finites only. It fucks things up at a fundamental level, and you need a certain adaptability in your assumptions to intellectually deal with the consequences of infinities.
That’s the only barrier to this discussion breaking out of this otherwise infinite loop.

This is convenient, huh? Simply define infinity as being able to have a completed “all” like finites can. Except infinity is the opposite of that, and can’t.

Let’s give Joe an infinitely large mouth - he both infinitely overshoots with his bite because of the infinity of his mouth, and infinitely undershoots due to there always being infinitely more apples.
Undefined.
It’s incapable of definably being completely changed or destroyed - any notion of a completed “all” is invalid.

I love magic stories!
And then what happened?! Feeling… sleepy… :sleeping-sleeping:

I meant to type 9 instead of 8 Silhouette, glad you caught it.

So here’s the deal.

Can you see the difference between value and correspondence?

Does PI not register infinitely more value than 3.000…?

I know they correspond, but what do you make of value?

I stand corrected.

Are you saying that my logic is invalid?

Are you saying that it’s not true that we can know that an infinite product of (0) is (0) if we know that (0) raised to any number (whether finite or infinite) is equal to (0)?

How about an infinite product such as (1 \times 1 \times 1 \times \cdots)? How do you know the result of this product is (1)? Is it because we know that (1) times any quantity (whether finite or infinite) is (1)? Or is it because we know that (1) raised to any quantity (whether finite or infinite) is equal to (1)?

That’s probably because you’re deeply insecure and have a strong need to see flaws in people around you in order to feel good about yourself. And you’re looking for any kind of flaws, so as long they are flaws – big or small, significant or insignificant, etc.

Normal people don’t do that.

It’s of no help if what you’re doing is looking for a number that does not exist e.g. a finite number that is equal to the result of the infinite product (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots). But if you’re merely trying to figure out whether such a number exists, then it’s quite a bit of help. It tells you that such a number does not exist.

The limit of an infinite product is not the same thing as its result. They are two different concepts.

It tells us that the result of the infinite product is smaller than every real number of the form (\frac{1}{10^n}) where (n \in N). Most importantly, it tells us that no matter how large (n) is, the result is always greater than (0).

Your argument is basically that there are no numbers greater than (0) but smaller than every number of the form (\frac{1}{10^n}) where (n \in N).

That’s one of our points of disagreement.

It does not merely “look like”. It will never ever get to zero for the simple reason that there is no number (n) greater than (0) that you can raise (\frac{1}{10}) to and get (0).

You can say that (0.\dot01) is approximately equal to (0), and that is true and noone disputes that, but that misses the point of this thread. We’re asking whether the two numbers are exactly equal not merely approximately equal.

You can say that (\frac{1}{0}) can be substituted with (\infty) for practical reasons (given that (0 \approx \frac{1}{\infty})) but you cannot say that (\frac{1}{0} = \infty) given that there is no number that you can multiply by (0) and get anything other than (0).

So from your point of view, the only conclusion that should make sense is that (0.\dot01) is a contradiction in terms, and thus, not equal to any quantity. By accepting such a conclusion, you’d have to agree that (0.\dot9 \neq 1). So at least one point of our disagreement (really, the main point of disagreement) would be resolved.

Still, one point of our disagreement would remain, and that would be your insistence that (0.\dot01) is a contradiction in terms based on the premise that there is no quantity that is greater than (0) but less than every number of the form (\frac{1}{10^n}, n \in N).

You have yet to explain where’s the contradiction.

Statement 1: “At some point in time at some point in space, there exists an infinite line of apples.”

Statement 2: “At some other point in time, no apples exist anywhere in space.”

How do the two statements contradict each other?

Certainly, the word “infinite” does not mean “not being able to be something else at some other point in time”.

…the exact point at which infinity becomes self-defining, so yes… anything infinite is not bounded within a defined measurable set.

A good definition… it’s not what you’ve got, it’s what you ain’t got. I like it. :smiley:

…but then wouldn’t that simply mean that something is either infinite or not? which I ‘think’ Silhouette (I don’t want to put words in his mouth) is also saying.

I am not exactly sure what you mean, so I’ll have to make a guess.

I suppose what you mean is that something is either finite or infinite i.e. that it cannot be both at the same time.

I agree with that. The number of elements within a set is either finite or it is infinite. There is no third option here.

Unfortunately, I cannot understand how that relates to what I said in the above quote.

The following two statements certainly do not make a claim that a line of apples existing at some point in time at some point in space is both finite and infinite.

They merely state that at one point in time the line is made out of an infinite number of apples and that at some other point in time the line is made out of zero apples.

As for Silhouette’s claim that:

It’s not true that you cannot make an infinite quantity bigger. As for the rest, it’s difficult to respond to because it’s difficult to understand.

See, the issue I have with you Magnus, is not that 0.666… is larger in value than 0.555…

The issue I take with you, is that there’s ANY infinity that is NOT in correspondence.

Neither of those numbers is infinite.

0.666… is exactly equal to 2/3

0.555… is exactly equal to 5/9

They’re just regular numbers.

You can easily confirm it by doing a long division. (A level of math which is taught in middle school.)