Is 1 = 0.999... ? Really?

in the case of (1,3,5,…,6,4,2) both ends of the sequence go toward infinite counting by twos, one ascending and one descending, nerd.

You agreed that (0.\dot9 \neq \infty). And it’s not a finite number. What’s left?

I’m somewhat familiar with the concept. I thought maybe that’s what you were getting at with your diagram:

(\bullet \bullet \bullet \cdots + \bullet \bullet \bullet \cdots)

^ Sort of a whole new set of numbers beyond the real numbers (“beyond infinity” so to speak), but we didn’t end up going there.

If this is what you’re talking about, then I think the inequality:

(\infty) + 1 > (\infty)

is simply poorly written. I would write it:

R + 1 > R

where R is a hyperreal number.

(\infty), as I said, is, at least to me, not a number but a description of a set: it’s infinitely large. It’s a property of the real numbers (and I would guess, of the hyperreals as well). But if you want to do arithmetic on “infinitely large numbers”, you could probably do it on hyperreals where such numbers are denoted R (or some variation). I wouldn’t use (\infty) 'cause that’s just confusing.

To me, hyperreals are just “numbers in a different universe” so to speak, so I don’t think there would be any problem doing arithmetic with them. This is sorta what I was getting at when I talked about “reaching infinity” as being like reaching the edge of the universe, going beyond, and in a manner of speaking, entering a different universe (this was pages and pages and pages ago). This is where I introduced the idea of speaking of infinity as a unit and never speaking of the infinite quantity it represents again. It’s because, once you enter this different universe, it’s as if the entities you left behind in the previous universe no longer exist relative to the new universe. This is just a fancy way of saying the reals are an infinite distance away from the hyperreals and therefore might as well be treated like they don’t exist. This is sort of why we call the normal numbers “real” and the ones beyond infinity “hyperreal” (I’m guessing mathematicians probably learned their lesson from calling (\sqrt{-1}) an “imaginary” number).

So far in this discussion, I haven’t granted the reality/validity of the hyperreals (and my research on the internet tells me not all mathematicians do either), but I will grant that if you really want to develop a mathematical system based on numbers “greater than infinity” you could probably do so without running into too many internal inconsistencies. But its relation to the standard mathematical system based on real number would have to be thought through very carefully. For example, does it make sense to mix real numbers and hyperreals in arithmetic? Or if you want to quantify an infinite set, which hyperreal number do you choose? And do hyperreals have their own infinities (i.e. an infinite distance on the hyperreal number line)? Is that what 2 x R would equal (R being a hyperreal number)? Or would it, like real numbers, just be another hyperreal number on the same number line that’s twice the distance as R is from the zero point on that number line?

I’m still not gonna grant that hyperreal numbers actually exist. I don’t believe that if the edge of the universe is an infinite distance away, the possibility of an abstract mathematical system of hyperreal numbers proves that you can add one more unit of distance to that infinite distance and step into another universe that’s a hyperreal distance away. Such a mathematical system would just be an abstract idea. But if we run with that idea in this discussion, it might be interesting to see what comes out of it. The first thing I would want to do is to agree on some ground rules–what can and what can’t be said about hyperreals–or at least, lay out my assumptions.

Yes, you can invent new answers. Just like Louis CK (jokingly) did when he said: “Well, because things that are not can’t be!” ← In his bit, this started from his daughter asking why she couldn’t go out and play in the rain. Do you really think before he said “no” he thought “things that are not can’t be… yada yada yada… therefore, my daughter can’t play in the rain.” The point is, you’ll be doing that ad nauseum if your opponent’s motive is to unearth the grounds from under your claims. He’ll just keep asking why until you’re not able to invent a satisfactory answer.

That’s a very noble motive, but in my experience, that’s very rarely what’s driving people in a heated debate. Most of the time, each party wants to destroy the other’s argument–it isn’t to share in the goal of finding the truth–which is why a lot of people will simply relentlessly demand justification upon justification upon justification… until their opponent is either exhausted or tongue tied or mistakenly says something that contradicts something else they said earlier (which is far more likely to happen if you’re pressed to invent justifications on the spot). It’s more of a game, a competition, than an honest and cooperative pursuit for the truth.

Right, you would simply say “it’s bigger” or “contains more objects”, etc. So you would reserve the word “infinite” to mean “endless” and you would agree it describes a property of a set with an infinite number of elements, and that it wouldn’t change by adding or removing elements from that set. But you do think the number of elements in the set increases by adding 1 to it. Would this be represent as R + 1 where R is a hyperreal number?

Sure, but only in the sense that rearranging the bricks in a building doesn’t mean the building hasn’t changed. But that’s irrelevant to comparing it to another building and asking: do they still contain the same number of bricks?

You could think of it like that, sure. But be careful with this: “Since I’m a god, I know what is true.” ← This might hold in terms of setting up the parameters that define the thought experiment, but it doesn’t mean you get to impose whatever conclusion you want. You can’t just say: “I’m a god and this is my universe, therefore if I say the number of apples increases when gib adds one, then the number of apples increases. QED!” Obviously, in that case, you could say whatever you want, and I can say: “That’s just an imaginary scenario that has nothing to do with reality.”

But what I mean when I say “I’m expecting to be able to detect a difference…” is not that I in the scenario expect to detect a difference (indeed, I couldn’t know), but I the thinker, the one imagining the scenario (I’m a fellow god too :slight_smile:), expect to detect a difference. I’m saying I need a way of conceptualizing the infinite set before the change as different from after the change. You’re saying: just imagine it as having one more apple. But then I say: that’s hard to do when the line is infinite. And I bring up the example of the line whose points were removed and the remaining points shift to take their place as an example. Even if we say the line now has fewer points, I can’t conceptualize that. I can’t tell (visually or abstractly) where there’s a difference between the one line and the other.

:laughing: Different quantities have different labels - does that mean you’re “relabelling” the same quantity? Of course not! Does it mean that a different label is the only difference? No!
Again!! The superficial consumes you and your thought, to yield superficial conclusions with no bearing on the reality that they represent.

Anything for you, Magnus! :romance-hearteyes:

The quantity of shits that I give is the difference between (1) and (0.\dot9)

No wonder the more I explain the less you understand - you think that reiteration to help you understand makes something less true and that it must be only emotionally driven.
The superficial has consumed you!

Sure, whatever you need buddy. I can be as sane or insane as you need me to be in your mind, and the logic of my arguments will stand.
For the 4th time, your judgments of me make zero difference to the undeniability of the reasoning that just happens to being delivered by me - but it could be anyone and it would be just as true.
The superficial is at your core, and you at the core of it.

Stop caring about your judgments of me - they don’t affect the irrefutable arguments laid before you.
The only person who matters here is you, and your learning to accept reason.

Again you’re putting words in my mouth that I’ve not said, to make a straw man for you to attack.

The closest thing to what you think I’ve said is that the set as a whole has a beginning (first element) and an end (last element), and so do the 3 specified elements at the start and those at the end.
Therefore since the number of elements has no end (as you clarified is meant by the ellipsis), it’s either impossible for the specified last 3 ending elements to ever come at the “end” of the “never-ending” ellipsis, and/or impossible for the ellipsis to ever even start if you’re specifiying an end to the ellipsis for the last 3 ending elements to be able to ever be arrived at and be able to continue from what’s in the ellipsis, and/or the ellipsis means a continuation of both progressions from both sides at once “inward”, in which case the endless ellipsis has all ends defined and is disappearing up its own arse with no “middle point” that can ever be reached to connect each side - making it two endless sets bolted together by their lack of end, which is a contradiction.

You pick which impossible interpretation you want and it’ll still be contradictory.

I’ll just wait for the penny to drop for you over here, like an insane person :banana-dance:

True and true.

Basically, what you’re saying is that we cannot reach the last position of the sequence ((1, 3, 5, \dotso, 6, 4, 2)) by starting at the first position (occupied by (1)) and then moving a finite number of positions to the right. That’s correct.

If you’re saying that this implies the sequence has no end, then I would strongly disagree. It merely means you cannot reach the last position the way you’re trying to reach it (i.e. by making a finite number of steps.)

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).