Right. Then you agree that it’s not a valid input to the doubling function, right? So your idea of (f(\infty)) is irrelevant since (\infty) is not a natural number. Yes?
Yes, the way you’ve defined (\infty) it’s the upper limit. I’ve agreed to that. The way you’re defining it is essentially what mathematicians call (\omega).
The set of real numbers between 0 and 1, inclusive, has 0 and 1 as its ends. What say you?
Another example: Take the naturals in their usual order but put 5 at the very end:
0, 1, 2, 3, 4, 6, 7, 8, 9, 10, …, 5
We implement this by defining “funny <” defined as n < m in their usual order, except that n < 5 for all n different than 5. Now this is an infinite set with a beginning and and end; a smallest and largest element.
“Endless” is a metaphysical definition perhaps, but the mathematical definition is as I gave it.
Could you please give an example of an infinite set that’s not Dedekind-infinite?
Why not? I challenge you to name a natural that’s not pair uniquely with an even; or an even that’s not paired uniquely with a natural. Please accept the challenge.
Aren’t you saying something without a shred of proof?
I’m interested in seeing you prove the negative.
I’m interested to the extent that you are willing to engage with what I’m saying. If not, not. But you do owe me two things:
A set you claim is infinite that is not Dedekind-infinite; and
A proof that there is a natural not mapped uniquely to an even, or an even not mapped uniquely to a natural, by the doubling function and its inverse.
Explain clearly how this correspondence between the naturals on the top row and the evens on the bottom row is not a bijection. Show me a natural on the top not mapped to a unique even on the bottom; or an even on the bottom not mapped to a unique even on the top.
0 1 2 3 4 5 6 ...
0 2 4 6 8 10 12 ...
Ok three things:
An admission that since you agree (\infty) is not a natural number, your idea of inputting it to the doubling function is inapplicable.
I said, repeated, and explained that by your claim that there was a bijection between the sets you implied that the limits were equal.
Just because the equation has the same form doesn’t mean that I am putting the variables in from the same set. I put in the limits which I explained were NOT in the number set. It was a different equation of the exact same form making a different equality of different variables (the limits - not the numbers).
So they use one symbol and I used a more common symbol but defined exactly what I meant by it. Why are you even saying this?
Irrelevant. The endlessness of the real numbers between any natural numbers is in the COUNT of those items. It is the count that is endless, not the “ends” defining the category. Even a single number can have an endless string of decimals used to representing it. It is the count of the items that is endless - infinite - not necessarily the defining ends of the category.
That is provably incorrect. Merely the number Pi has an infinite, non-repeating string of decimals constituting an infinite set that has no equinumerous subset. There are many others - a random set, decimals of ε, and others. We can’t say they are not infinite just because they are not Dedekind-infinite (requiring an equinumerous subset).
I do not need to prove the negative to assess that you have not proven the positive. The specific reason your string of example bijections is not a proof is that you have not presented the case that ALL numbers would end up being paired - that the limits really are equal. Apparently to you that seems a trivial and obvious issue (I thought mathematicians were very sensitive about that). But it is not. And it is where the fallacy resides. A proof requires every detail to be elucidated - otherwise it is just an inference.
I sense a degree of disingenuousness (in fact a little hostility).
On this board when most people take a stand, if they are the slightest hostile, regardless of any evidence, they simply deny the evidence or reasoning or they just ignore it. I have explained one of these issues 3 times and you seem to simply ignore what I say and reclaim your objection - “disingenuous”. I have tried to be cooperative in hopes and expectations that we could gain progress and see eye to eye. I feel like you aren’t going to allow that regardless of anything I say.
If you are still wondering about the need to equate the limits in order to prove your theory let me explain -
If I merely say that there is a bijection between two infinite sets by starting an example set and claiming that it continues infinitely, I could claim the following -
A = {0,1,2,3…}
B = {0,2,4,6,…}
“And there is a perfect bijection between them” (your theory).
But also I could claim -
A = {0,1,2,3…}
B = {a,&,R,@,4,-2,-1,0,1,2,3…} “And there is a perfect bijection between them”
I could list any set that ends with any infinite set as a subset and claim a perfect bijection to those as well because there is no evidence that every element has been paired.
That is why we have to prove that the limits are identical as well as a perfect bijection.
Not at all. I neither said nor implied anything about “limits,” which are inapplicable and irrelevant in this context. We have two sets, the naturals {0, 1, 2, 3, 4, …} and the evens, {0, 2, 4, 6, 8, …}, and an obvious bijection between them given by f(n) = 2n.
0 2 3 4 5 6 ...
0 4 6 8 10 12 ...
Limits have nothing to do with this. I wonder why you think they do.
To point out that there’s already a standard notation for your idea, and that (\infty) isn’t it.
If I accept that, what of my other example 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, …5? This is a simple rearrangement of the natural numbers that has both a smallest and largest element. It’s a very common example in set theory in fact. It has the same cardinality as the naturals, (\aleph_0), but a different ordinal, (\omega + 1) as opposed to (\omega). It falsifies your idea that an infinite set has no end or is endless. (\omega + 1) does in fact have a beginning and an end, in quantity AND IN COUNT.
All of those sets are Dedekind-infinite. In the case of the decimal digits of pi, just correspond the set of all the digits with the set of the even-indexed digits via the 2n mapping.
Of course I have. n gets paired with 2n and 2n is paired with n.
That’s irrelevant and meaningless. Limits are not a factor in this context at all.
I have done so.
This I don’t see. You said, “I can easily prove the negative,” or words to that effect. I didn’t look up the exact quote. And I said, “I’m interested in seeing you prove the negative.” I still am. In order to prove that f(n) = 2n is not a bijection you have to show me a specific natural not mapped to a unique even or a unique even not mapped to a specific natural. You need to do that in order to falsify my claim that 2n is a bijection. You said you can do it, I asked you to do it.
Project much? Seems to me this is exactly how you are operating.
Here are some links to discussions of Galileo’s paradox, which dates from 1634 evidently and not 1538 as I mistakenly said. Since this idea has been accepted for over a thousand years at this point (going back to the ancient Greeks and the Arabs of the 1200’s at least) the burden is on you to explain what you know that Galileo and others didn’t. That’s not hostility, it’s a statement of observable fact that you are denying something commonly accepted to be true.
I not only said there is a bijection, I identified a specific bijection, namely f(n) = 2n. One that is very familiar to anyone who made it through high school Algebra 1. That’s ninth or tenth grade in the US, 14 or 15 year olds.
Well actually there is, since you’ve only added finitely many new objects to a countably infinite set.
It’s easy enough to do. a goes to 0. & goes to 1. R goes to 2. @ goes to 3. 4 goes to 4. -2 goes to 5. -1 goes to 6. And then 0 goes to 7, 1 goes to 8, 2 goes to 9, and in general n goes to n + 7 from now on. You can verify that this is a bijection. Each element on the top pairs uniquely with an element on the bottom, and each element on the bottom pairs uniquely with one on the top.
Please note that I have not only claimed a bijection, I have demonstrated a specific bijection. That’s how you show two sets to be in bijective correspondence. You are required to demonstrate a bijection or, at the very least, demonstrate that some bijection must exist.
The example you’ve just given is Hilbert’s hotel with seven new guests arriving at an infinite hotel that’s already full. You just shift each existing guest in room n to room n + 7, freeing up 7 empty rooms for the new guests.
[Quibble with myself: You already listed 4 so I don’t have to pair up 4 the second time it appears, since sets can’t contain duplicates. Unless your ‘4’ was a new symbol that happens to look just like the usual symbol for the number 4. But no matter. Adding finitely many elements to an infinite set never changes the set’s cardinality).
Limits are irrelevant and inapplicable here. What you need to do to prove a bijection is supply a specific bijection as I have done here in each case. That’s the only way to prove a bijection: Identify a specific function that is provably a one-to-one correspondence between the two sets of interest.
By God this is exactly right. .999… = 1 is a theorem in mathematics. It’s “true” by virtue of being a valid theorem if you accept the axioms of math; and only by virtue of that.
As such, it is no more controversial than asking if the knight “really” moves that way. The question is a category error. The knight moves that way because that’s one of the rules of the formal game of chess. And .999… - 1 is a consequence of the rules of the formal game of math.
The reason people get confused about this is because they try to fit it into physics or metaphysics, and those two are category errors. .999… = 1 is not true in physics because we can’t measure anything that precisely due to measurement imprecision in general, and the Planck length in particular, which says we can’t measure anything smaller than (1.6 \times 10^{-35}) meters. So past the 36th or so decimal place we can’t say anything sensible about the matter.
And when you try to throw in metaphysics, it’s hopeless. Infinity is endless, boundless, divine, ineffable, God-like, whatever. None of those descriptions have mathematical relevance. Infinity in math has a particular technical meaning. People get in trouble trying to apply vague philosophical ideas to precise mathematical terms.
As far as Leibniz, well I’m more of a Newton fan but yes I’ll give Leibniz his due. And old Archimedes had an excellent grasp of infinitary processes over a couple of thousand years ago.
So that is the defense for your theory - “If we ignore it - it doesn’t exist”?
To me that would be like -
[list]Me: 17+2 = 37
Yu: I don’t believe that
Me: It’s true
Yu: Prove it
Me: It’s obvious
Yu: Show me your math
Me: maths have nothing to do with it. It’s just true.
Yu:
[/list:u]
So you also believe that paradoxes are real? Interesting.
Let’s say that N = the natural number set = {0,1,2,3,4,5…}
And that B is N reordered to {0,1,3,4,5…,2}.
It is easy to see that N⤖B because both sets are equality countable = equinumerous.
It seems to me that you’re putting words in my mouth here. I said nothing of the sort.
A bijection is a function. It’s a mapping of the elements of one set to the elements of another set. Limits aren’t involved. I don’t have to justify that, any more than if you claimed grapefruits are involved. Limits aren’t involved in the definition of functions. That’s a mathematical fact.
I’d add that the idea I’m expressing – not “my” theory at all, but a basic part of mathematics for over 140 years now with roots going back thousands – is standard and universally accepted. Of course that doesn’t in and of itself make it correct. But it does place a higher burden on you if you want to claim that everyone else is wrong about there being a bijection between the naturals and the evens, and continue to maintain your objection in the face of this graphical representation of such a bijection.
0 1 2 3 4 ...
0 2 4 6 8 ...
This picture seems convincing to me; and I do not understand your insistence that limits are somehow part of the definition of a function, when a high school student can tell you otherwise.
I have not ever in this thread, going back to my earliest participation in it, said that anything is “just true” because I say so. I always give solid mathematical reasons. I have done so here. I have defined what a bijection is and demonstrated a particular one between the naturals and the evens.
In order for you to claim f(n) = 2n is not a bijection between these two sets you would need to demonstrate a natural not associated with a unique even or vice versa; and this you have not done.
Is that what you think I said?
I mentioned Galileo’s paradox because it shows that the idea of bijecting the natural numbers to one of its proper subsets goes back hundreds of years and was noted by one of the greatest scientific thinkers of all time. I did pose to you the question of what you think you know that Galileo didn’t.
I don’t know what the symbol ⤖ means in this context, but I would agree that these two sets are equinumerous, one being a permutation of the other.
Assuming that ⤖ means “equinumerous with,” no they are not, because the set {N, a} has cardinality 2, as can be plainly seen; whereas N has cardinality (\aleph_0); and (2 \neq \aleph_0).
Yes, its cardinality is 2, and 2 is countable (though not countably infinite). Sometimes “countable” means countably infinite as opposed to finite, so it would depend on which sense of countable you’re using. But the cardinality of C is 2. It’s a set that contains two elements, N and a, whatever a is supposed to be.
Cardinalities don’t drill down. The cardinality of the set {The Mormon Tabernacle Choir} is exactly 1, even though the Mormon Tabernacle Choir currently contains 360 members. Taken as a whole, there is only one choir; and the set containing it has cardinality 1.
But perhaps I’m misunderstanding your intent. Did you mean to add a to N; that is, take the union of N with the singleton set {a}? That’s countably infinite, in the same way I showed you a bijection between the natural numbers and the natural numbers augmented with &, a, and so forth that you asked about earlier. It’s exactly the same question.
I’d like to ask you a question I’m curious about, going back to how we got onto this bit about the naturals and the evens. You said infA is the number (or degree if you like) of the set of natural numbers; and infB is the number or degree of the set of even natural numbers. What is the inf degree of the multiples of 3, of the multiples of 4, 5, 6, and in general the multiples of n? What about the primes, or Galileo’s example of the perfect squares? How about all the other categories of natural numbers? There are after all uncountably many subsets of the natural numbers. Does each of them get its own inf-level? Did James ever address that question? Have you given it any thought?
How about sets that are larger than the naturals? The integers, for example, or the rationals, or the reals, or the complex numbers or the quaternions? Do they also have inf-levels?
I’m asking because you and I got onto this conversation about the even numbers when you defined infB as the degree of the even numbers and I pointed out that whatever that means, the cardinality of the evens is the same as the cardinality of the naturals. I’m still trying to understand exactly how infA differs from infB. What quality or aspect of the evens is captured by infB such that it’s clearly different than infA, and then what about all the other possible subsets and supersets of the naturals?
Can you explain your reasoning for this? I think .999… = 1. But even if someone disbelieves that, they are going to say that .999… is just a smidge less than 1. People tend to think it’s “smaller than 1 by an infinitesimal amount.”
1 is finite so a quantity a tiny tiny bit less than 1 is still finite. I don’t follow why you think this is infinite.
“countable” does not mean having to increase the cardinality else every set is countable.
James explained that the use of his “inf(Variable inserted)” is an arbitrary choice made during a calculation. The variable “A” was merely his most common example - “infA = 1+1+1+…”. Any variable can be chosen. InfB can be any other infinite set of interest. They are not fixed labels for predetermined sets.
I chose that infB, during this discussion, was representing the even naturals. I could have chosen infQ.
For sake of keeping it clear, I suggest not using the word “infinite” in this discussion and replace it with “endless”. And when discussing the “very end of the endless series or sum” perhaps call that “infinity”.
And as wtf explained, we normally do not refer to the endless length of decimals of a number but rather the value those decimals represent. The value of 0.999… is very nearly 1.0 - a very far distance from being infinity (almost the exact opposite). As I tried to explain earlier - each of those right-of decimal digits gets increasingly smaller and when added we have an infinite sum of infinitesimal amounts which finally sums to a finite value.
The idea that a finite in the physical universe cannot ever become infinite is not actually true. Take this case -
In a small area in space there exists only 2 infinitesimal items (whatever they might be or be called). If it is also true that speeding toward that area are an infinity (infA) of other similar items that just happen to reach that area at the same time - viola - suddenly those 2 items are joined by an infinity of others.
So we go from merely 2 to infA+2 in merely moments.
James used that little jewel to explain how it is that particles, having an extreme mass density ever get formed. He had a pictorial of such an event converging at a single point in space -
I think his MCR = Maximum Change Rate. He was explaining why waves of affectance (energy waves) delay to the point of becoming stable particles of mass. With an infinity of converging waves from literally all directions of even an infinitesimal average value, an infinite value must be summed at the convergent point. But because the single value cannot grow from finite to infinite (as you stated), a delay in wave propagation occurs that ends up growing and creating a traffic jam of affectance known as a “subatomic particle” - made sense to me.
But if we are just talking about the total count within an area, that number can easily grow from a finite number to infinity under the right circumstances (no value summation at a single point required).
Gottfried Wilhelm Leibniz invented and founded the differential and infinitesimal calculus (1665) seven years earlier than Newton (1672), which suggests that Newton stole from Leibniz. Leibniz invented a multiplying machine (1673), Newton the reflecting telescope (1669). Newton also founded the laws of gravitation (1666).
Leibniz’ causal principle and final principle, his theorem of the reason and his differential and infinitesimal calculus led in the application to physical processes to the interpretation of the laws of nature as extreme principles (differential, integral or variational principles), the binary number system with the digits 0 and 1 (dual system) developed by him to the computer technology which he already initiated with his constructed calculating machine. Even more: Leibniz had an effect in all fields of knowledge and on all fields of science.
In the public it is always said: Leibniz invented and founded the differential and infinitesimal calculus 7 years earlier than Newton. Newton would have done it independently of Leibniz, but this is not true, because Newton stole from Leibniz.
Invention/foundation of the differential and infinitesimal calculus: 1665 (Leibniz) and 1672 (Newton). So Leibniz was the inventor/founder.
Publication of the differential and infinitesimal calculus: 1684 (Leibniz) and 1687 (Newton). So Leibniz was also the first publisher.
That’s right. But the “equation” works mathematically, so mathematically said it is an equation (without quotation marks); but the “equation” does not work philosophically,so philosophically said it is either no equation or an “equation” (with quotation marks). Thus the problem is a philosophical one but not a mathematical one. And that means that it is discussable as a philosophical but not as a mathematical phenomenon. It works mathematically. It works!
Leibniz can still be felt everywhere today. In mathematics, in physics, in technology in general, in science in general, and not least in philosophy.
Actually, I didn’t want to address this rather general topic at all, but only address the topic of this thread (“Is 1 = 0.999… ?”) and just say that Leibniz, to solve a problem, used a trick, which was ingenious. This has had effects until today and will have effects also in the future, unless we will forget all science.
