Is 1 = 0.999... ? Really?

I forgot my other argument:

If you add a 1 at the front, you have bumped the entire infinity up 1 space, which is exactly the same as adding a 1 at the end (both of which means the series wasn’t really infinite to begin with)

I’m sure I have more, but these three should suffice.

Every partial product of (0 \times 0 \times 0 \times \cdots) is equal to (0) and that’s precisely why its result is equal to (0).

The same does not apply to (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots).

Who told you it’s impossible? I said it’s unnecessary.

It’s because you’re a confused and at the same time conceited mind (hence your defensiveness.)

I accepted that (A) and (B) are identical, not that (B) and (B’) are identical.

Yes, it will be good for you. If anything, you’ll be less focused on who you are in relation to other people and more focused on the actual subject.

Can you do this for us, please?

Or do you REALLY have to tell us how great you are each time someone says something negative about you?

(0.\dot01) is a number that is greater than (0) but smaller than every number of the form (\frac{1}{10^n}) where (n \in N).

That’s what (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots) means. It is telling you “This is a number greater than (0) but less than (0.1) and (0.01) and (0.001) and (0.0001) and so on”.

You can say “But there is no real number that can represent such a quantity!” to which I can respond with “But the set of real numbers is not the set of all quantities.” I can then add that numbers are merely symbols invented by humans in order to represent quantities that are of relevance to them. They are not quantities themselves. Symbols are few and far between, quantities are numerous. And when it comes to quantities, they can be as small or as big as you want them to be. There is no limit.

All in all, all you have to do is understand what that infinite product represents. But you’re refusing to do so, obviously because you have your own agenda – it’s in your interest to misinterpret it.

What’s certain is that (0.\dot01) is not a number that is less than itself. That’s your own invention.

A contradiction would be a statement such as “A number that is greater than (1) but less than (\frac{1}{2})”. That would be a contradiction because we said that a number is greater than (1) (P) and not greater than (1) (not P.)

No such contradiction is committed when we say that (0.\dot01) is a number that is greater than (0) but smaller than every number of the form (\frac{1}{10^n}) where (n \in N). That’s because every number that we say is greater than (0.\dot01) is also greater than (0).

Another contradiction would be a statement such as “A number that is greater than (1) but less than (1)”. A number can’t be both greater than and less than one and the same value. But that’s not what we’re doing either.

Of course, it is possible that we are contradicting ourselves in other ways, by contradicting other statements that we’ve previously made, but the fact is, there are no other statements we’ve made.

The problem with Silhouette is that he’s starting with the premise that there are no numbers smaller than all real numbers. If you START with such a premise, and then you say there is such a number, that would be a logical contradiction. But noone other than him (and those who think like him) start with such a premise.

Certainly not me.

And since this is MATHEMATICS and not EMPIRICAL SCIENCE the only thing we are interested in is LOGICAL CONSISTENCY. You can’t say that some premises are true and others are not. The only thing that matters is consistency: making sure you are not contradicting what you previously stated.

Why would anyone start with a premise that there are no numbers smaller than every real number?

There is LITERALLY no limit to how small a number can be (in your head, of course.) No matter how small a number is, you can always imagine a smaller number. There is no limit whatsoever. All it takes is a statement such as “Here’s a new number N that is smaller than every number that is currently familiar to me!” You don’t have to imagine it or visualize it. All it takes is to state its relation to other numbers.

So what’s the point of limiting yourself to real numbers?

Inability to think abstractly?

Nothing, because I explicitly said that “Most of my objections are actually logical ones, and occasionally I back them up with empirical objections - either way, in order for you to forward your conceptual propositions as knowledge, they need to be testable and tested.”

I never said only empirical claims are testable - you are making things up AGAIN.

Unfortunately you don’t even realise that you have been implying that this thread is about (your) untestable claims at multiple points. Every time you claim you don’t “need” to prove your (il)logical claims, or that the logical arguments of others either don’t exist when they clearly do, or you simply assert they’re false when you notice them whilst at the same time accusing others of that very same fallacy that you’re committing, as well as every time empirical tests are offered to help you understand your errors that you simply claim are irrelevant - you’re arbitrarily invalidating the possibility for both logical and the empirical disproofs to apply to your attempted “proofs”.

Basically you’re constructing epistemological double standards for this thread to make it impossible for validity to prove you wrong whilst simultaneously dictating that invalidity can prove you right. In itself this is fallacious - “moving the goalposts”. This is why you have no intellectual honesty, integrity or authenticity.

Homework for you: devise an argument that would successfully cause you to change your mind that isn’t simply a reiteration of your own points and how you think they’re right.
That is intellectual honesty - the ability to evaluate both sides of the story: the existence of which I’ve been trying to communicate to you all this time.
Obviously you’ve not been in higher education, or you’d be versed in this kind of thinking because that’s what qualifies you for it and qualifies you to succeed in it. Either that or you completely disregard this kind of thinking especially for the internet - but from what I’ve seen I’m going to have to guess you never even made it, and perhaps realised it was never worth you trying. Must be the fault of the institutions though, right? Never yours.

You’re correct that I intentionally add an extra (0) to (B).
You’re incorrect that I’m “relabelling” the elements of set (A). I’m adding (1) to each of their elements and appending the result to the (0) that I first append to (B’). The fact that this makes them identical to (B) is no accident - this was intentional to show you that alternative set construction to end up with the same infinite set doesn’t objectively change the set even if it now “looks like” injection when in fact set identity ensures bijection.

Obviously my efforts were yet again wasted.

I use the term “simplify” in the technical sense of making absolutely no change to the essence of the contents, not colloquially such that it potentially implies concealment or trickery. My terminology is wasted on you.

(B’) has all the elements that (A) has, with 1 added to all of them before they populate (B’) - to the preliminary ends of ({1, 2, 3, \dotso})
That’s why I add (0) first so that this set starting from 1 is appended to a set that has (0) already prepopulated within it.
I defined (B’) as ({0, 1, 2, 3, \dotso}) - stop pretending I didn’t and actually read my explanation. No trick.

The only “trick” is if you mistake bijection for injection because of the intentionally alternative appearance of (B’)‘s construction compared to (B) (and (A)) to conclude that (B’) now has more elements when if you follow each corresponding element to infinity it’s literally impossible to find any injection.

I wish it could be, but unfortunately I’m waiting on you. I won’t give up until you’ve learned something: you the self-confessed non-mathematician professing mathematical expertise.

What would you most like to take from this thread? Victory? Self-improvement? Reconciliation? Teaching your expertise? Name it.

I can repeat myself too if you want.

That’s not getting us anywhere though is it.

I know every partial product is greater than 0. This just panders to your appeal to finitude that you keep denying you’re doing.
INFINITES. We’re dealing with them, not “partial, finitely somewhere along the way of an infinite series” which you’re pretending still counts as valid for evaluating an infinite series.

I showed you that it’s impossible to narrow down any possible value above 0 to fill in any “gap” between (1) and (0.\dot9)
The only value that cannot be divided smaller, thus satisfying this “gap” is 0, otherwise it can always be made smaller.
I’m showing you that it’s literally impossible. You’re just saying it’s unnecessary for it to be possible, but you’re calling it “unnecessary to prove an exact amount”.
The problem is that you don’t appreciate the implications of your vagueness.
This is mathematics, not speculation.

You’re picking up that I’m defending logic and reason huh? Sharp!
Defending myself though? Fuck that - I don’t give a shit what anyone thinks of me unless it suffices as a means to the end of enabling the grander scheme of rationality to be upheld. I don’t want that goal to elevate me or bring me down, it has nothing to do with me - rationality is objective and has nothing to do with the person who forwards it. It works regardless, independently. The only personal satisfaction that I take from philosophical discussion is in the aesthetics of truth. It’s nice. Obviously you don’t think so, but that’s a result of your personal issues and has nothing to do with me personally. Call this kind of dissociation from conceit “conceit” all you like. I give zero shits.

(B’) is defined to biject perfectly with (B), just with alternative construction to make it “look like” it’s different to people fooled by the superficial.
As I keep saying and proving, that’s you.

I’m not giving up on you, Max.
No matter how frustrated it makes me.
I will make you learn even though I’ve been taking the entirely wrong approach of being honest with you about yourself this entire time. I should have been softer and less confrontational - I already knew that this just makes people act irrationally in exactly the same way that you are. I did the same thing with Urwrongx1000 but after a couple of years even he realised I was onto something.

I’m willing to wait that long for you, Magnus.

Oof! If you meant N as (\Bbb{N}) as the set of natural numbers you’re in danger of starting to sound like you’ve been studying up on the subject!

Now we just need to eliminate your reliance on the invalid term (0.\dot0{1}) and maybe we can get somewhere?!

Perhaps if we formulate it like “Given the quantity (x) such that (x=1-0.\dot9), (0<x<\frac1{10^n},\forall{n}) where (n\in\Bbb{N})” we might start sounding remotely close to professional.
But I’m just a mathematician, I’m not a professional one, so if anyone wants to correct me with any legitimacy - please feel free. I’m pretty sure there’s a more succinct and correct, purely symbolic way of legitimately expressing that.

But this is just notation - the argument remains that whilst (\forall{n}) where (n\in\Bbb{N}), (\frac1{10^n}-0\gt0), (\prod_{x=1}^\infty\frac1{10_x}\not\gt0)

Basically “partial” products don’t cut it by definition - like we already covered.

I know what the expanded product of your infinite product is intended to represent.
I know that “there is no real number that can represent such a quantity” - as you say.
It is therefore your duty to comprehensively a valid population of quantities that satisfy the ability to exist between (0) and the difference between (1-0.\dot9)
You need to stop cowering from the formalisation of such a proof, apologising that it’s “not necessary” by virtue that it “ought” to follow from what you regard to be a valid argument, which I have proven to be logically invalid.
You need to do better.
The Hyperreals already attempted this way before you, but as I’ve argued their conflation of the infinite with the finite is only a useful tool rather than truth.

It’s not enough to say that “there should be some quantity there”, you need to validly justify it - my “agenda” is with intellectual rigor ONLY. I don’t give a rat’s ass about anyone’s excuses to get around that - I will hunt down your mistake and destroy it. It’s fun and I enjoy the aesthetics of the result, but I don’t care one jot what form it ultimately takes - even if it turns out you’re ultimately right somehow.

It’s a fact that (0.\dot0{1}) is a contradiction regardless of the fact that it was me who identified it. You cannot logically declare an infinite series of zeroes entirely bounded by a decimal point and a (1) because entirely bounded contradicts the boundless quantity of zeroes in between. Simple. Logic.

Exactly!

Ok, I see. That would make sense considering you don’t agree that (0.\dot9) = 1. But I didn’t know you thought it doesn’t even equal a quantity. I thought, for sure whatever Magnus thinks (0.\dot9) does equal, he thinks it’s a number. But I take it you think that because the infinite sum of 0.9 + 0.09 + 0.009 + … never attains 1 (or just never completes), it doesn’t even attain the status of being a number. Is that right?

I always thought that so long as it falls somewhere on the number line, it’s a number:

… ← -3 – -2 – -1 – 0 – 1 – 2 – 3 → …
…^ ← (0.\dot9) falls here (either 1 or just before 1).

Infinity obviously falls outside the range of the number line (what I mean by “beyond numbers”) and so it does not fall on the number line, and therefore isn’t a number.

It’s true that when a series approaches a limit, that limit can sometimes be attained and sometimes not, and the question of what kind of limit does (0.\dot9) approach is up for debate, but I always thought that the proof given at the wikipedia article settled the matter. If not, what would?

Because arithmetic deals with numbers and only numbers (which are, by definition, finite). It is the practice of calculating numbers. Throw anything else at it–like red, or cows, or shoes–and it makes no sense.

You’re calling the assertion that arithmetic only deals with finite numbers “popular opinion”, huh? I’d call it self-evident, pretty close to axiomatic–that’s just what arithmetic is.

You can ask me to justify my claims ad nausium, but we’re getting pretty close to elementary principles here, which is to say rock bottom, end of the line. Soon it begins to be like asking: why does 1 + 1 = 2? Little children do this. The incessant chain of never-ending whys. Louis C K has a good bit on this:

[youtube]http://www.youtube.com/watch?v=Tf17rFDjMZw[/youtube]

The fact is, no one has an infinite chain of justifications stored up in their minds for all the things they believe (you’re no exception). Press someone to justify their claims long enough, and you’ll eventually get to the end of the line. That doesn’t mean they don’t know what they’re talking about, it just means you’ve reach the fundamental level on which what they believe is based. At this point, it’s just intuitive, self-evident, axiomatic. It’s like trying to explain red to a man color blind from birth. If you find you can’t do it, that doesn’t mean you don’t know what red is, it just means your knowledge of it is fundamental, basic, doesn’t break down any further. At this point, you just know it or you don’t.

So you agree that endlessness is a property of sets, one that sets either have or they don’t. Do you agree that the word “infinite” is just another word for endlessness?

When you say you make an infinite set bigger by adding a new member to it, would you describe this as “more infinite”?

When you say (\infty) + 1 > (\infty), you obviously don’t mean “more endless” since you just agreed that endlessness is a property that a set either has or doesn’t have–no degrees–and I’ll await your reply to see what you say about “more infinite”–but could (\infty) above be replaced with [the size of an infinite set] + 1 > [the size of an infinite set]?

I can see how it might seem intuitive that adding a member to an infinite set would increase the overall number of members in that set, but because the number of members of the set is directly tied into its property of being infinite/endless (in fact, defining it as such), that for me brings into question what things we can say about its size (number of members) and what things we can’t say.

The point of the parallel lines thought experiment I brought up was to show how when you remove a subset of points from one of the lines, making it appear to have changed, rearranging the remaining points can show that nothing has really changed in the line at all. It reacquires its prior state exactly. I know we had a disagreement about whether it was really the same or there was a persistent change, but I explained that it’s not about which points are which but that the structure remains the same. I used the building analogy to solidify this point (you didn’t reply to that which lead me to wonder whether you read or not, so let me know if this doesn’t ring a bell). The overall point is that if we want to say that something about an infinite set changes when we add or remove elements from it–whether that’s its size, length, weight, density, whatever–I’m expecting to be able to detect a difference between the set’s before and after states. But that’s the problem with infinity. Infinite sets turn out to be very resistant to change, at least in terms of their quantity. Assuming uniformity of all its members, the things in it can be rearranged and shuffled around such as to assume exactly the same structure after adding or removing elements as before. And for this reason, I don’t find it as easy as you to simply say the size of the set increases/decreases by adding/removing members.

Not really. I didn’t say it’s not a number. I said that it does not evaluate to a finite number.

That’s what I call being fooled by the appearances. The number line represents the set of real numbers and the set of real numbers is not the set of all numbers that can be imagined – it’s merely a small subset of it.

Have you heard of the hyperreal number line? That’s a different kind of number line. It not only includes infinity, it also includes different degrees of infinite and infinitesimal quantities.

They aren’t finite by definition. I’m not aware of any such definition. It’s just that we are mostly familiar with finite numbers (since they are of the greater relevance to us.)

Quantities have to do with more/less. Anything that can be said to be less than (or more than) something else is a quantity.

Every “Why?” question can be answered provided that you’re interested in doing so. If you aren’t, you don’t need to justify yourself. You can simply leave the discussion.

The point is to examine each other’s assumptions and see who’s at the wrong side. But again – you don’t have to do this if you don’t want to.

Yes.

That’s not how I would call it.

But rearranging the remaining points does not show that the line didn’t change. It merely makes it look like it didn’t change. It’s a magic trick.

I did but I didn’t get to respond to it.

When we say that you should imagine yourself standing in front of an infinite line of green apples, what we’re doing is creating an imaginary universe in which you participate.

The creators of this universe are its gods (which is really only me) and you are one of its participants. Since I’m a god, I know what is true. And since you’re not, you may or may not know what is true.

One of the gods of this universe proclaims “There is a man called Gib standing in front of an infinite line of immovable and indestructible green apples. One day, the gods gift him with a red apple and he decides to place it in front of the green ones.” The question here is whether it follows from this description, i.e. whether the gods are making a statement, that the line is longer than it was before.

The gods may continue and say “This man called Gib, as well as every other man that shall ever exist, will never be able to know whether any two infinite lines are equal in length or not.” That does not mean that the gods didn’t say that the line became longer after addition, and also, it does not contradict that statement because what is and what can be known are two different things.

The only thing we want to do here is understand the kind of universe that I’ve just described.

Well it’s not an infinite number. Having an infinite number of digits in a base-10 representation does not make a number infinite.

“Infinite” is being used to refer to the number of digits in the representation of a number, the number of terms in a series expansion, the index of a term in a series, the number of elements in a set, the index of a element in a set, the sum of a series and the value/size of a number.

It’s no wonder that this thread is a confusing mess of mixed up concepts, ideas and results.

It’s certainly less than (1 + 1 + 1 + \cdots). But it’s not finite either.

Yes.

Magnus, if you add a 1 at the beginning or end, you are adding a new dimension, not increasing the number…

You know, like IP addresses!

234.564.108.22

Those dots stand for different dimensions…

Yes, you’re adding (1) to each of their elements, and by doing so, you’re relabelling them.

If you have to say it, you do.

Maybe you should say it one more time?
Just to be sure?

You are officially insane.

An infinite sequence such as ((1, 3, 5, \dotso, 6, 4, 2)) is not a logical contradiction even though its bounded from two sides. This is because the ellipsis is not telling you that the sequence has no beginning (first element) and no end (last element.) It’s merely telling you that the number of elements has no end.

I’d be more than happy to continue this discussion with you, but first, you’d have to stop being so hysterical.

Yes, there is a logical contradiction. 1,3,5 … (say infinite 1’s) is a rational number - no problem there!

Tacking 6,4,2 at the end is not how math works.

there’s an infinite amount of divisibles between any two numbers yo
you can do (0…1) and that is an infinite set
that is how math works

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.