If you don’t show why they are not then you’re right - your words will not mean anything.
I’ll leave you two to it. I did my civic duty here. The burden is on you to define those terms, since they are mentioned nowhere else in math or science other than on this forum. How could I refute something that’s not even coherently defined? There’s an obvious bijection between the even whole numbers and the entire set of whole numbers; on the one hand; and between the multiples of 4 and the whole numbers, on the other. So you could never argue for any sensible cardinality-based distinction between the evens and the naturals.
So your argument is authoritative rule? Do you also believe in Global authoritarianism (which is what you now have) - Truth reigns from above - independent thinking not allowed?
I don’t believe that. I don’t believe that you can show an “obvious” 1to1 bijection of the naturals and the even naturals.
Yeah ok man. f(n) = 2n. Have a nice day. This is going to be pointless. Like I say, I’ll leave you two alone.
Ok I think I see why you think what you think. Let’s unpack what it is to be a ‘…’
Consider this:
2 4 6 8 10 12…m versus 1 2 3 4 5 6 7 8 9…m
Replace ‘…’ with ‘inf’ and then consider this:
A consists of 12345678inf
B consists of 2468inf
I cannot say A is semantically half the size of B just because the digits before the inf/… are all even, or that there are twice as many digits before the inf. What I can say is this: In terms of symbolic/digit representation, A is half the size of B. Again 12345678…m and 2468…m both denote the same measure. One label/number/symbolic representation is half the size of the other. That is all that is half the size.
Do you agree with the above bold and underlined?
NOT “well-defined”.
I thought you required that?
If your intention was to do your civic duty, then you have my thanks. I am also trying to do mine.
Ok I’ll play.
What about f(n) = 2n is not well-defined in your opinion? We match the natural numbers to the even natural numbers as follows:
0 - 0
1 - 2
2 - 4
3 - 6
and so forth. Each natural goes to a unique even; and each even comes from a unique natural. What is your issue with that?
Galileo made the same observation in 1538 or so regarding the perfect squares 0, 1, 4, 9, 16, 25 … being equinumerous with the whole numbers.
en.wikipedia.org/wiki/Galileo%27s_paradox
And even though Galileo got his name on it, the same observation was made in the 1100’s or so by Arab mathematicians. So what exactly is your concern? Please be specific, I’m wondering what it is about this simple demonstration that troubles you.
I was disappointed that after I explained that expressions like 888… or 999… don’t have any meaning, you continued using them; and that after I explained that .999… m or km is meaningless in physics, you kept on using it. And that after I noted that .999… = 1 refers to pure numbers and not to anything in the real world, you complained that .999… = 1 cake isn’t true – something that I had already agreed to. I did get the distinct impression that you didn’t read anything I wrote. especially about the cake. I agree with you that .999… = 1 is not true about cakes. It’s true about the real numbers.
You still did not define your “inf” - exactly how many iterations are represented - what degree of infinite?
Realize that if you say that -
x = 111… and
y = 222…
You could add 1 to either and not know any difference - not know which one you added to - because your “…” or “inf” hasn’t been specified exactly.
Let me explain what I thought was a useful notation that James came up with -
If we define the natural number set quantity as infA we can note an infinitesimal as -
1 / infA = [0.000… …001]
That is just his way of noting that it is not zero but infinitely close to zero.
Then if we add 1 infinitesimal to 1.0 we get -
[1.000… …001]
And that is well defined and easily distinguished from merely 1.000… No information is lost.
If we do not note it that way, our 1.000… + 1 infinitesimal would not be distinguishable from 1.000… The infinitesimal disappears due to the notation.
The same is true for infinite values.
If we add 1 to 111… we get -
111…
So where did the additional 1 go? - it is just not shown - and that leads to a many confusions and paradoxical proofs. It is just a matter of the notation not being sufficiently defined. That is how they prove that 1=2 or that 0.999… = 1.0
James would have noted it as something like -
[111… …112]
just so the logic details turn out perfectly with no paradoxes available.
So using James’ notation concerning the 1 = .999… issue we see it as -
1 = [1.000… …000]
0.999… = [0.999… …999]
1 infinitesimal = [0.000… …001]
And
[0.999… …999] + [0.000… …001] = [1.000… …000]
thereby proving that you have to add 1 infinitesimal to 0.999… in order for it to equal 1.0
Irrelevant.
They are just symbols I used to pick a particular semantic I had in mind, but I tried to understand you regarding this, so what I actually did after was ask you the following:
Ok, for the sake of argument, when discussing Infinity, forget numbers. Do the following words have any semantical value: ‘Infinitesimal’ ‘Infinite’
I don’t know where you stand on this. Does ‘infinite’ and/or ‘infinitesimal’ have any semantical value for you? If so, then any number sequence with ‘…’ is my way of picking out those semantics.
Again, I am not in disagreement without our inability to empirically observe the Infinite/Infinitesimal. We don’t have the tools for this. But I am in disagreement with suggestions that they have no semantical value. I am also in disagreement with suggestions that they have no relevance to our Existence. This is because we cannot accept Existence being finite as it logically implies Existence coming from non-existence.
At the time it seemed like you were just changing the subject. Since 888… and 999… have no meaning, it’s pointless to ask questions about them. And since you subsequently used this undefined and meaningless notation, it did seem as though you did not read what I wrote.
The infinite definitely has mathematical meaning as the transfinite ordinal and cardinal numbers; that is, Cantor’s work and its modern continuations. Infinitesimals don’t exist in the real numbers, but they can be implemented in the hyperreal numbers of nonstandard analysis. But these are very different subjects than the way you’re using the terms. For example in nonstandard analysis there are lots and lots of infinitesimals, not just one. And they do not in any way shape or form operate as James’s fallacious infA concepts.
Well, Existence is a much broader philosophical topic, on which I take no position. I’m only discussing the mathematics of .999… = 1. I don’t think it necessarily means anything outside of pure math. It’s a statement about the real numbers; and the real numbers arguably have no physical existence at all.
A good analogy is chess. We wouldn’t ask if the knight “really” moves that way, or what it means about existence. It doesn’t mean anything. It’s a rule in a formal game. Likewise .999… = 1 is a logical consequence of the rules of the formal game of mathematics. That’s the position I take. I don’t try to figure out what it means, because it doesn’t really mean anything unless you want it to, as for example using infinite series to solve some differential equation to solve a scientific problem. Even then it’s just a calculational device, not a statement about the world or existence or anything else metaphysical.
Iterations of what? Can you clarify for me what you mean by “inf”?
I don’t think you can add to x or y. You say my ‘inf’ hasn’t been specified exactly. Consider the following variations in specifications:
How do you add length to a road that is 111…m or 222…km long? Which road is longer? The one that is 111…m or 222…km? Or is there something in the ‘…’ that will determine which road is longer than the other? Can you tell me what that is?
I read what you wrote. And I don’t think I rushed in reading it. Our intentions are different.
I am trying to have a non-paradoxical understanding of the semantic of Existence. This is impossible without the semantic of Infinity. I agree that there is only 1 Existence. But I think It Infinite. Which is why earlier on I said .999… = 1 Infinity but not 1 pie. I did not mean to say that you said 1 pie. I just wanted you be clear on what I’m mean when I say I agree that .999… = 1
So if you think .999… to be meaningful, why do you not think 999… to be meaningful too? Don’t they mean the exact same thing?
The notation 888… has no meaning in math. You keep using it without defining it. To do so is meaningless.
A worthy endeavor to be sure. I wonder why you’re doing so in a thread on .999… = 1, which frankly has nothing to do with existence.
Ok. Contemporary physics doesn’t have a theory of infinite existence, but that’s only a contingent fact and could change next week. But what does this have to do with .999… = 1?
A point with which I wholeheartedly agree.
When I say that .999… = 1, I mean that this is a valid theorem in the modern theory of the mathematical real numbers. I do NOT think the statement is meaningful in physics or in the real world, because we can never measure anything that precisely. We can’t physically sum an infinite series.
I consider it to be mathematically meaningful but not physically meaningful.
Because there’s no mathematical interpretation or definition of that notation.
No. The notation .999… is a shorthand for the infinite series ( \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \dots ), which is shown in freshman calculus to be a geometric series with sum exactly equal to 1.
The latter (999…) is a notation that you have invented but not defined, that does NOT have any mathematical meaning until you give it one, which you have not done.
As you have already agreed earlier, all natural number expressions are finite in length: like 6, 47, 112, or 43504385904380584385. There is no mathematical meaning assigned to an infinite string of digits without a decimal point in front. So unless you tell me how you intend such a string to be interpreted, it is meaningless.
You are correct, from a moral point I suppose, that there’s a bit of an asymetry. An infinite string of digits is meaningless without the decimal point in front; but becomes meaningful when there is a decimal point in front. I agree that may be somewhat unintuitive, but it’s just the way it is.
wtf, your words on mathematics are the most respectable on this forum.
I have an advanced mathematical education but it seems you’re still the only one here who seems to have studied it in any great depth.
As such I’m always glad when you weigh in on the subject.
I think if anyone can speak sense in thread like this, it is you - so I can wish you good luck, though I know you have the odds stacked against you with the current crowd.
CR, ur relatively new here, but I’ve noticed some strange things that you’ve been saying on this thread - if you want to make use of someone who knows what they’re talking about, then I recommend this guy as a valuable resource.
There might be no textbook that speaks of (999\dotso) and no mathematician who has ever worked with such a symbol but that does not mean that the meaning of that symbol cannot be logically deduced from existing definitions.
In fact, I would say that it can.
(999\dotso) stands for (9 \times 10^{\infty} + 9 \times 10^{\infty - 1} + 9 \times 10^{\infty - 2} + \dotso).
I read the entire thread and I don’t really think you ever had much of an argument. Perhaps you should present one now? Note that what you’re doing here is merely asserting that you think that “infA” and “infB” are non-sensical, not well defined and not logically consistent. You said what you believe but you did not say why you believe that.
Exactly.
I would say they are well defined. “infA” stands for “the number of natural numbers”. That’s how James defined it and that’s also how Observer defines in the post that you responded to. As for “infB”, it means “the number of even natural numbers”. It was also defined in the post that you responded to.
But if these definitions aren’t good enough for you, I’d suggest explaining what’s unclear so that a better definition can be provided.
I don’t agree with that.
If there is a bijective function between two finite sets, they are necessarily equal in size, right? There is no bijective function between two finite sets that are not equal in size.
Consider (A = {1, 2, 3}) and (B = {4, 5, 6}).
1 → 4
2 → 5
3 → 6
When you pair every element in (A) with a unique element in (B), no unpaired elements are left in (A) and (B).
But also, if you can map every element in (A) to a unique element in (B) without mapping every element in (B), they are necessarily unequal in size, right?
Consider (A = {1, 2, 3}) and (B = {4, 5}).
1 → 4
2 → 6
3 → (none)
When you pair every element in (A) with a unqiue element in (B), you necessarily end up with an unpaired element in (A). If you can do this with two finite sets, the two finite sets are necessarily unequal, right?
Now let’s consider the set of natural numbers and the set of even natural numbers.
There appears to be a bijective function between the two of them:
1 → 2
2 → 4
3 → 6
etc
But also, there appears to be the following injective non-surjective function between the two of them:
1 → (none)
2 → 2
3 → (none)
4 → 4
5 → (none)
6 → 6
etc
This suggests that the two sets are both equal and unequal (a logical contradiction.) But for some reason, you choose to ignore that, instead merely focusing on the fact there appears to be a bijective function between the two of them. But the fact is that by using your very own logic, I can “show” that the two infinite sets are unqual in size.
How do you deal with that?
(\sum_{i=0}^{i=>\infty}(9 * 10^i) = 999…)
(\sum_{i=0}^{i=>\infty}(8 * 10^i) = 888…)
But I guess sophisticated mathematicians couldn’t figure that out. 
Actually, that’s not it.
(999\dotso \neq 9 \times 10^0 + 9 \times 10^1 + 9 \times 10^2 + \dotso)
(999\dotso = 9 \times 10^{\infty - 1} + 9 \times 10^{\infty - 2} + 9 \times 10^{\infty - 3} + \dotso)
But the bottom line is that it can be logically deduced from existing definitions. There’s no need to invent new definitions (which is what wtf’s arguments boil down to.)
How did you “logically deduce” that the exponents start at infinity-1?
And where do they end? … you don’t say … they could go on endlessly past infinity-infinity to infinity-infinity -1, infinity-infinity-2, …
At least Obs has a series except for his weird summation upper bound.
I’m sure the order used to depict a serial addition is totally irrelevant in maths (so your first inequation is false) - and also that infinity-1 is definitely a non-starter - “go beyond the end of the endless then step back one” - sounds like something to tell your annoying mother-in-law.
Yes. That was supposed to be an arrow not a equal-or-greater sign - 
-
(\sum_{i=1}^{i->\infty}) – since i is never to be infinity itself
But
(\sum_{i=1}^{\infty}) – will normally do