Is 1 = 0.999... ? Really?

All bijections equal each other.

Infinity is an operator; a function.

There are no greater or lesser infinities.

I don’t know where you get that. It seems senseless to me.

Infinity is not a quantity. There can be no lesser or greater than.

It’s just as much an operator or function as the plus sign is. It takes any or all the other operators and just adds to it “forever”. That’s all infinity is. Not a quantity. An operator / function

I read you the first time. Repeating it doesn’t explain it (I hope I’m not sorry I asked).

Ok, so you understood it in that language the first time. Give me your argument why it’s false then.

Would that not depend on whether or not we ascribe time to them? Let’s call these somethings As. There is area x and area y. At time t1, there are no As in y. At time t2, As begin to enter y. There will never be an infinity of As in y. As far as I can see, we either remove time from this equation, or, we accept that As cannot go from x to y. They were always in y (but this has its own problems if x is viewed as an existing/real location)

There is one confusion I have with the above:

Either ‘free space’ is that item that is between any and every identified item, or ‘free space’ is an item that is sustained/encompassed by that item that is between any and every identified item. I don’t know where you stand on this.

Let’s discuss the item that is between any and every identified items. I don’t think you can have two such items because that would be like having 2 sets/items of all items. If 1 Item is between any and every identified items, then there cannot be another Item that does this, right?

If my objections above hold, then we cannot interpret “high peaks of affectance” as being infinitesimals. This does not mean that we have to deny their realness or existing. We just have to interpret them as something else.

Yes, I think we are getting closer and closer to identifying where we differ and why we differ. God Willing, we will get to the truth regarding this matter.

I did ascribe time to the spikes by saying that they were “headed toward” the area.

Why did you conclude that?
If there were an infinity of them (and I am convinced that in the real universe there really are) and all headed in that direction, how is it that they will never get there? :confused:

It is like a wave on water. The sea is filled with water and the wave is an item moving across that water. Between one wave and another wave (2 waves) there is water. And at the same time the waves are made of that water. The waves are items identified as different than the surrounding water (because of their peaks). Now just change the word “water” with “affectance” and you have the scenario I was talking about - two “items” (waves/spikes) traveling within a small area of affectance (the “sea” - or in this case an actual area of space).

Can’t you have a set of numbers where two of the numbers are unique from the others yet they are ALL nothing but numbers? The items (the 2s) are made of the substance (numbers).
{1,1,1,2,1,1,2,1,1,1,1,1,…}
I am not seeing any problem with this. And those 2s represent the spikes within the infinite sea of nothing but numbers.

And why can’t they be regarded as infinitesimal spikes?

You are asking me to prove to you that flying is not an aeroplane. All I can think to tell you is to look it up in the dictionary. The action is not the item. (\infty) is not an action/instruction/process/function. It is an item - a noun not a verb.

If you are ascribing time to this matter, and you accept that you cannot count to infinity, and you view the spikes as being separated from one another, then I think there are semantical inconsistencies with what you are proposing because of the following: An infinite number of finite time measures would have to pass by for what you are describing to be accomplished: For every separate spike to go from x to y, a finite amount of time needs to pass by (call this amount of time A). You cannot start/begin from 1A and then get to InfinityA. If the spikes were not separated from each other, then the spikes would not be spikes. They would be what I think you call Affectance/Sea/Water and I call Existence. Infinite time/space is only true of the Omnipresent (Existence). You need Infinite time for what you describe. You cannot have it when the spikes are separated.

I will try to convey to you my understand using your analogy: I think the water/sea to be infinitesimals absolutely united as opposed to divided/separated. This literally makes/means/amounts to 1 Infinity of infinitesimals, or, 1 Infinite Item/Thing. Because if one infinitesimal is not separated from another infinitesimal by something other than infinitesimal, then it is all just one Item/Thing consisting of an Infinity of infinitesimals. The spikes cannot be infinitesimals purely because semantically, nothing can separate one infinitesimal from another. Also, I hesitate in saying infinitesimals because if x is not separated from x by anything other than x (in other words, if x is separated by nothing), then it’s just one x.

I’m not sure I understand what you mean here. To me, what you suggest amounts to like having a tally system, except where I = 1 and III = 3, you are saying we can have two different symbols that = 1/I. But again, I’m really not sure what you’re trying to convey here.

Are infinitesimals separated by Affectance? If yes, then doesn’t that imply that Affectance can be smaller than an infinitesimal? How else would it separate one infinitesimal from another?

I think you have to view Affectance/Sea as an Infinity of infinitesimals. You have to view the spikes as being something other than infinitesimals. If you don’t do this, then I think what you propose would amount to the following: Nothing is truly omnipresent. Nothing encompasses all things (or no one thing encompasses/sustains all things). Only when it comes to Existence can you say i am in It and It Is in me without being contradictory (despite appearing to be (I know how you feel about the set of all sets being a member of itself)).

Did you miss the part about all of those spikes moving at the same time, in parallel, as a group inward from the surrounding area. It would be no different than a handful of sand being thrown into a pail.

That would be the sea - all of the infinitesimal portions connected.

That seems to make no sense at all. We can’t have an infinitesimal spike of water on one planet and another on another planet?

Remember “infinitesimal” is a measure of an item, not an item in itself.

So sorry for the late reply. I only just saw your post now.

Are we in agreement that you cannot count to infinity? I will assume yes. With that in mind, there are two ways I interpret 0.999…

A) There is no end to the numbers/fractions. B) The numbers/fractions continue on forever

There is a clear difference between A and B (despite it not looking that way).

To me, B implies that .999… is converging/continuing on forever towards 1 without ever reaching it. Whereas A is the number Infinity for the following reason:

B does not mean an infinity of fractions/numbers. B means counting to or continuing towards an Infinity of fractions (which cannot be done so B cannot consist of an infinity of fractions). A on there other hand means an infinity of numbers/fractions. Thus:

Does it not stand to reason and semantics that if 0.999… implies an infinity of fractions/numbers, then it is implying the quantity/measure of Infinity/Infinite/Infinitesimal? It does not mean smaller and smaller fractions being added continuously such that the sum total forever converges towards 1 (it is not B), nor does it mean converging towards infinity/infinitesimal, it means Infinity/Infinitesimal/Infinite. For example: An infinity of 1s = Infinity in quantity. An infinity of 1 cms or 1 kms or 77 ms, all denote the only thing that is Infinite in length (Existence). An infinity of natural numbers or fractional numbers either = the quantity Infinity, the measure Infinite, or the measure Infinitesimal.

Again, if I keep going with the following 1, 1, 1, 1, 1, I will never reach and infinity of 1s. But If I say an infinity of 1s, or an infinity of any things (fractions, letters, numbers, digits) then I am highlighting the quantity Infinity.

That is how I understand it.

Which of the following do you mean:

  1. The infinitesimal portions are connected by the sea.
  2. The sea = All of the infinitesimal portions connected

If you mean 2, what is connecting the infinitesimal portions?

I agree. But as far as I can see, there is only one item that can have that measure.

Does it matter how you write it down? You can do it any way you want. There isn’t much of a difference between your approach and my approach.

Wikipedians do it the same way that I do except they start with (0).

And the reason we do it this way is because it’s more elegant.

Consider this (your approach):

(d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^0)

And now consider this (Wikipedia approach):

(d_n \dotso d_2d_1d_0 = d_n \times 10^n + \cdots + d_2 \times 10^2 + d_1 \times 10^1 + d_0 \times 10^0)

Which of the two expressions look more elegent to you?

And this isn’t the only reason I write it that way. There’s another one.

As for why I start with (1) rather than (0), there’s a reason for that too.

Right. If (\infty) stands for “any number greater than every integer” then a number cannot be both greater than every integer and at the same time less than any number greater than every integer. That would be an obvious contradiction.

That’s why, as you already know, we have to do one of the following things:

  1. we have to say that, in our case, neither (\infty) nor (\dotso) indicate “any number greater than every integer”, but rather, that both indicate one and the same specific number greater than every integer

  2. we have to come up with a neutral symbol, such as for example (x), which we can say represents some specific number greater than every integer; and we also have to say that the number of digits in (999\dotso) is (x) (to make it clear, we can use something like (999\dotso_x))

Let’s pick the second way because it’s less likely to lead to confusion.

My claim is that (999\dotso_x) stands for (9 \times 10^{x-1} + 9 \times 10^{x-2} + 9 \times 10^{x-3} + \cdots) where (x) stands for some specific number greater than every integer.

I also claim that 1) YOUR expression and MY expression represent two different numbers, 2) my expression represents a larger number, 3) my expression represents (999\dotso_x), and 4) your expression represents (\dotso999_x).

Now, let me take the following statement of mine . . .

. . . and reword it, like so:

The non-zero digits in (999\dotso_x) are digits whose index is less than (x) but greater than every integer.

But I think I made a mistake here.

Let’s see.

The index of the first digit is (1), so the index of the last digit is equal to the number of digits. Since the number of digits is (x), the index of the last digit is (x). Thus, the non-zero digits in (999\dotso_x) are digits whose index is less than (x + 1) but greater than every integer.

Do you agree so far?

(999\dotso) is a decimal representation with the following properties:

  1. it has no fractional part (no decimal point is explicitly stated and the ellipsis does not imply it)

  2. the number of digits is greater than every number (this is implied by the ellipsis)

  3. it has the most significant digit but it has no the least significant digit (this is implied by the fact that the representation does not end with a digit)

  4. every digit is (9)

(999\dotso) is not a number, but rather, a category of numbers. This is because there are two things about it that are not fully specified. These are: the number of digits and the place of the most significant digit.

The number of digits is partially specified because we know it’s greater than every integer. The place of the most significant digit is also partially specified because we know it cannot be a place associated with an exponent that is less than the number of digits minus one.

phyllo appears to be right in that the exponent associated with the first digit in an instance of (999\dotso) is not necessarily the number of digits minus one. It can be the number of digits, or the number of digits plus one, or the number of digits plus two and so on.

Just a general note -

I can understand, although disagree with, wtf’s argument. I understand what he is trying to say and claim - and why.

But when it comes to Certainly real, Magus Anderson, and Ecmandu - you blokes seem to be making some really weird arguments that seem to be just arbitrarily scrambled thinking - like “why not just throw in this confusion - or maybe this ambiguous notion - ah, here’s a conflation that might make things more cloudy…” :confused:

I just have to hope that it isn’t intentional.

What I see is a distinction without a difference.
But actually in A there are no fractions.
And in B there are only fractions (or actually decimals).

    • unless I have them backwards from your intent. :confused:

That part is right (except the use of the word “Infinity” rather than “an infinity”).

A = the big one – 999…
B = the small one – 0.999…

  • that’s what I guess as how you mean them

No, I don’t think so. “0.999…” has nothing at all to do with counting. And it does imply the sum of an infinity of decimals (or fractions). But it is still only one decimal number - that sum, not the list of decimals.

That misunderstanding is one that I understand - I see the confusion (for that one).

B (assumed to be the 0.999…) is a single number. It is NOT a series of actions. The representation of “…” does not mean that anyone is to actually add forever but rather that IF anyone tried to complete the decimal, they would have to add forever (so don’t try).

B is the final sum IF all of the decimals were added regardless of the fact that no one could actually add them. We can know the sum anyway. We can know that something is or isn’t infinite without going and counting it. We can logically deduce a result. We don’t have to always calculate it.

Again, “999…” is a single number. There is no implication of having to add anything unless you just want to try it. And if you add ALL of the infinite parts, the sum of that addition would be the single number “999…”.

And all the above is why this -

  • makes no sense.

I guess you mean this edit:

Is this the edit you meant?

I can agree with it, although there is a a little problem with finding.

Finding is similar to the epistemological subjevtivity/objectivity problem, a duality. But nevertheless I agree to the said sentence “earnestly seek and ye shall find” and the followig sentence.

What you said about truth and reality - “you can hide from one but never escape the other” is even more agreeable, regardless whether I would use the word “facts” instead of the word “reality” as well. Both refer to what we perceive, namely what has become, that is, what is completed through history. So, I would use the word “facts” as well as the word “reality”.

It is right waht you said about truth and reality: “you can hide from one but never escape the other”. If you try to escape from truth, you have to face the reality (facts); if you try to escape from reality (facts), you have to face the truth. You are not as fast as the both are. Lies have short legs; unreality is like a dream in which movement is impossible.

There is no connector. They are merely side by side portions - like the right and left sides of a piece of paper. How one infinitesimal is distinguished from another is merely by which portion you are focused on or talking about. The distinction is an imaginary line drawn between to portions of an otherwise continuous substance - the North portion of the Sea or the South portion of the Sea.

Every portion (an item in our discussing of the whole of the substance) can have that measure (assuming infinitesimal portions). I don’t know where you get that there can be only one. How many infinitesimal portions are there of a pint of beer?

Obsrvr,

The only reason motion happens in the cosmos is because of infinity: infinity = motion (because it never ends - by definition)

You’re the person here who looks at dictionaries and can’t actually think to this regard, not me.

Just think about it.

Yes.

Actually I meant the reality of the future consequences due to the reality of the past events. If I say, “facts”, I just throw in more opportunity of get into the semantics of “who’s facts”. :confused:

And even though the following seems off topic here, it is actually related (to why bother pursuing these words and what they mean).

When a society tries to govern by establishing an artificial truth narrative (such as real world Communism and those trying to govern the US right now), they create an artificially induced public bubble of belief and force the issue of having to stop all thinking. Reasoning tends to unveil the reality that it is all artificial - creating doubt in the veracity of the narrative.

I think establishing an artificial reality narrative for a population can be good or bad but is most certainly a very, very dangerous thing (as has been proven over thousands of years of attempts). It stems from the lust to advance by trying to be God - in place of God (in place of Reality). And they actually know that it is extremely dangerous - to everyone else. They fully know that they are going to murder billions of people - they simply don’t care. To them being a god over all life is just too blindingly important. And they know that you can’t catch them. So they proceed.

Those kind of people seem to think that they actually have no choice. They think that the only way to advance is through the means they currently see - their “path to godhood”. And the reason they think that is because of what we are discussing on this very thread -

Getting the real story straight before making presumptions about what is necessarily true - look CLOSER at the details of your own thoughts and language (which largely affects your thoughts). Seek out, not merely A path to victory, but the BEST path at all possible (assuming you are not in a rush).

The problem is that it is an ethical issue. And when people, high or low, simply don’t care - they don’t bother (relating to James’ PHT). I think that on high they are very willing to murder billions of people because they haven’t bothered to seek out a more ethical solution to advancing humanity toward a more harmonious existence on Earth (another seriously big issue with James - “Saint”).

I can see where he is coming from. I started by merely being curious - then impressed - then amazed - then awestruck - then just speechless (as you said - “impressed with him” was only in the beginning). I have been closely examining his proposals and what I call his “final solution” (his SAM Co-op) for the possibility of any tiny little misguiding hidden devil. I haven’t concluded that his final solution is pure and more importantly that it is the BEST solution to human harmony. I think that I have just recently realized who he was talking to all that time (a serious question I have held) - and it was not anything I was guessing. His solution seems to be a calculated far off destiny - and AFTER billions of people have been murdered - exterminated by what he referred to as the source of all of the trouble - the blind lust of the “Godwannabes”.

I have nothing better to do besides my wife, work, and wealth pursuits than to see if his unfortunately distant solution, possibly after the death of literally all human life, is really the BEST possible. And that requires a degree of examination for that elusive devil in the details that a “normal” person ( :blush: ) would never bother to find.

In short - details matter in thinking and trying to solve real world problems involving humanity - not merely maths or scientific war weaponry - “Get the words straight” :smiley: