I agree that we can’t physically sum an infinite series. My reasons for saying Existence or Reality is Infinite, was purely because it is contradictory to believe that Existence or Reality could have come from non-existence. This logically implies Reality or Existence has always existed. I am now trying to argue that nothing other than Existence can be Infinite, because there is no room, or logical space available for there to be another Existence or Infinite thing. By doing this, I may be able to show that Infinity does not come in various sizes.
0 is not a meaningful measure (because it is not a measure at all. No measure = 0 measure). Infinity is a meaningful measure: If x has no beginning and no end, then x’s measure is infinity. I agree that we cannot empirically measure this, but it is still a meaningful measure (unlike 0). With this in mind:
I agree with your understanding that .999… is equal to 1 infinity. I understand this because between 0.9 mm and 1 cm, I understand there to be one infinite measure, and (but not plus)many finite measures. This is not the same as saying infinity + finite. This is the same as saying 1 infinity encompasses an infinite number of finites. What I don’t see, is how an infinity of measures = 1 cm. Or how infinity = 1 cm. I am not saying that this is what you said, I am just saying what I believe cannot be. And I say this because it leads to the following:
Infinity is between 0.9 mm and 1 cm. It is also between 0.99 mm and 1 cm. It is also between 1 cm and 10 km. It is also between 0.9 mm and 10 km (as in there are an infinite number of finite measures between each of the aforementioned measures). Infinity does not change in intensity or potency or size. There isn’t a bigger infinity between 1 cm and 10 cm than there is between 1 cm and 10 km. Infinity is consistently the exact same measure that encompasses an infinite number of finite measures (of which consists of 1 cm, 10 km, and all that is between 1 cm and 10 km, and all that is less than them or more than them)
So, if x is a finite measure, then it is fully encompassed by infinity, such that between it (x) and any other finite measure y, there is infinity (which in turn makes an infinity of finite measures between x and y possible. BUT It does not mean that Infinity is confined to just between x and y. Infinity absolutely encompassing x and y (such that it (infinity) encompassesbeing between x and y), is not the same as infinity being EXCLUSIVELY between x and y). x and y can be any finite measure. The same Infinity is between them and encompassing them. This infinity does not vary in potency/intensity or size. Nor is it in any way equal to any of the finite measures it fully encompasses. It encompasses as well as divides/separates one finite measure from another finite measure, infinitely (because it is Infinity).
Given your expertise, do you see any problems with the above?
What I am trying to say is, just as between every existing thing is Existence, between every existing thing is Infinity. Infinity and Existence denote the same thing. There is no beginning or end to them. There is nothing between Existence and Existence other than Existence Itself. There is only 1 Existence. Similarly, there is nothing between Infinity and Infinity other than Infinity Itself. There is only 1 Infinity. There is no 0/non-existence separating one thing from another. There is Infinity/Existence. I hope Obsrvr also sees the part in red.
To be mathematically correct, I should mention that, completely off-topic from the present discussion, there is a mathematical system in which such expressions make sense; namely, the p-adic numbers. The point being that you can define any mathematical notation you like, if you can make sense of it. Reading ahead, I would say that you have not succeeded in making sense of the notation as you’re using it. But I’m a little confused, it was @Clearly real who was using that notation. Are you endorsing it as well?
You already have a problem here. (9 \times 10^{\infty}) is not defined. And (\infty - 1) is not defined. You’ll have to nail those down first.
That is very impressive, my hat is off to you. Seriously. It’s way more than I’ve done.
Against infA and infB, you’re probably right. As I recall, by the time I got to the thread James had already been using these terms for a long time in other threads. I made an effort but never tracked their definitions back to the beginning. James posted mostly on physics topics that were far from my interests.
I do recall that he tried to link them to the hyperreal numbers of nonstandard analysis; and that his understanding of those numbers was flawed and incorrect. I do remember correcting him on many aspects of that. But on the definitions of infA and infB, you’re right, I never did get to the bottom of what he was talking about.
Fair enough. Let’s start fresh then, because the origin of those terms is not to be found in this thread, but is in other threads that I’ve never even seen. And James was nothing if not prolific. I’m willing to start anew, regretting that James isn’t around to give his own interpretation. So I’ll have to take your interpretation as official.
Ok. Let’s start here then.
Well, “number” is not defined in this context. All we really know about is the finite natural numbers 1, 2, 3, and so forth. Sometimes including 0 but that’s not important here. So the “number” of fingers on your hand is 5. We all agree on that.
In order to talk about “how many” elements an infinite set has, we have to define what we mean by that. The “number of natural numbers” is not defined till we say what we mean.
I will accept that you DEFINE infA as the number of natural numbers. This is the same as what mathematicians define as (\aleph_0), the cardinality of the set of natural numbers. Another name for this set is (\omega), the Greek lower-case letter omega. It’s the same set as (\aleph_0), but regarded as an ordered set.
The distinction is illustrated as follows. Suppose I reorder the natural number to put 5 at the very end, like this:
1, 2, 3, 4, 6, 7, 8, 9, …, 5.
To do this we would just define a “funny <” relation so that a < b whenever neither are 5; but n < 5 for all n except for n = 5.
Then the set {1, 2, 3, 4, 6, 7, 8, 9, …, 5} has the same cardinality as (\aleph_0); after all, it’s exactly the same set of elements, just rearranged.
But as an ordinal, {1, 2, 3, 4, 6, 7, 8, 9, …, 5} has both a smallest and largest element; whereas in the usual order (1, 2, 3, 4…}, (\omega) had a smallest but not a largest element. So we can change the ordinal number of a set without changing its cardinal.
In fact the ordered set {1, 2, 3, 4, 6, 7, 8, 9, …, 5} has the ordinal number (\omega + 1).
I mention all this because math already has a well-developed theory of infinite quanties and orders. You can make up a different theory if you like, but you have work to do.
Ok well here you are going to have some trouble, and unlike your use of “number of natural numbers,” which is a minor issue, here you have a major one.
The problem is that the cardinality of the set of evens is the same as the set of naturals. That’s because:
We DEFINE two sets to have the same cardinality if there exists a bijection between them.
In this case there is a bijection between the naturals and events, given by (f(n) = 2n).
Note that I am not saying that the “number” is the same, because I haven’t defined that. All I’ve said is that they have the same cardinality. This is important, because all I’ve done is give a definition. It’s not right, it’s not wrong, it’s not good, it’s not bad. It’s just a definition. We define two sets as having the same cardinality if there’s a bijection between them. We note that there’s a bijection between the evens and the naturals; hence by definition these two sets have the same cardinality.
Ok glad to do so. The problem is this: Suppose you show me a set of size (may I use that word, or do you regard that as incorrect?) infB. I say to myself, Ok, that’s a set having the same size as the even numbers.
But now suppose I had a set of children’s blocks of size infB. So I have blocks labeled 2, 4, 6, 8, 10, … and so forth. But then I grab some paint and from each block I erase the number written on it, and replace it with n/2. So now I have blocks labeled 1, 2, 3, 4, 5, 6, … Voilà! I see that I actually had a set of size infA after all! So in fact I just convinced myself that infA and infB denote sets of exactly the same cardinality.
This is exactly why I find the notion of infB superflous and not well thought out. Because it’s no different than a set of size infA, which is to say a set of cardinality (\aleph_0), as it’s called in standard math.
So I’m willing to say infA is coherent. It’s just another name for (\aleph_0). But now infB is just another name for (\aleph_0) as well. I see no difference.
To get ahead of some common concerns and objections here, let me note a couple of things:
In some sense we can say that the evens are “smaller” than the naturals, because the evens are a proper subset of the naturals. But this is a tricky business, because (f(n) = 4n) is a bijection between the naturals and the multiples of 4. And the multiples of 4 are a proper subset of the evens! So you have your “smaller” set of evens properly contaning an even smaller set of multiples of 4, yet the multiples of 4 provably have the same cardinality as the naturals. So again, if you like I won’t use the word incoherent, which sounds pejorative. Rather, I’ll just say that you still have some work to do to make your idea coherent.
There is in fact a perfectly sensible way that you can say that the evens are exactly half the size of the naturals; and that is the idea of asymptotic density, or what Wiki calls natural density.
To calculate natural density, we take the percentage of evens that occur in any initial segement of the naturals; and we note that as the number of natural increases without bound, the percentage approaches 1/2. This is very convenient in many applications. Likewise the natural density of the multiples of 3 is 1/3, and the natural density of the multiples of 4 is 1/4.
It’s not perfect, for example the natural density of the primes is zero. But still, it’s better than nothing.
So you see mathematicians CAN make such fine distinctions and use them as appropriate in a given context. But I don’t think this is what you are referring to.
How can you not agree with the fact that THERE EXISTS a bijection between the naturals and the evens?
Of course it’s true that there also exist functions between these sets that are not bijections; but so what?
You can’t deny that there’s at least one.
There IS a bijection between them. You can’t deny that.
I perfectly well agree that there are functions between them that are not bijections. But remember we are just applying a DEFINITION. We say two sets have the “same cardinality” if there is a bijection between them. That’s it.
This is a bit like a guy convicted of bank robbery, and identified in the newspaper as a bank robber. He sues the newspaper claiming that although it’s true that he robbed a bank once last week, he DIDN’T rob a bank every OTHER day of the year.
It makes no difference. If you rob even a single bank, even if you lived all your life and never robbed any other banks, you are a bank robber. It’s an existential (“there exists”) and not a universal (“for all”) quantification.
It’s perfectly true that some functions between the evens and naturals are not bijections. But at least one function is a bijection, and that’s enough to satisfy the definition.
It’s a definition. It’s not right or wrong, good or bad. If a thing matches the definition, that’s it. You don’t have to like it, it’s just a definition.
I hope I just did. The definition is that IF you ever robbed a bank, you are a bank robber. It matters not that on every other day of your life you did not rob a bank. If you ever robbed even a single bank once, you are defined as a bank robber.
If there exists a bijection between two sets, they are DEFINED to have the same cardinality. I said nothing else. You can’t disagree that there is at least one bijection.
Of course I do agree with you that there are lots of other functions between the naturals and the evens that are not bijections. That’s perfectly obvious. But so what? The definition says that if THERE EXISTS a bijection, the cardinalities are defined to be the same. The guy is a bank robber even though he’s lived for 15,000 days and on 14,999 of them he did not rob a bank. The definition is that if THERE EXISTS a day on which he robbed a bank, he is defined as a bank robber. If THERE EXISTS a single lonely pathetic little bijection between two sets, those sets are DEFINED to have the same cardinality. You can’t argue with a definition.
All you’ve done is to write down a pair of divergent infinite series. They don’t have well-defined sums. And you haven’t troubled yourself to attempt to define their sums.
Secondly your notation is weird. It would be more clear to you f you wrote (\displaystyle \sum_{n \in \mathbb N} a_n). That shows that you are summing (a_1) plus (a_2) plus (a_3) etc. You are summing over all natural numbers. That’s all the notation means. You’re trying to say it means something it doesn’t mean.
I don’t see what this has to do with math. I don’t think I can be of any service to you. I can help you clarify your understanding of math. But Existence with a capital ‘E’, that’s above my pay grade.
There’s something in math called measure theory. Clearly you’re not talking about that. I have no idea what you mean by measure. Why isn’t 0 a meaningful measure? The measure of the volume of purple flying elephants is zero. That seems clear enough to me.
In mathematical measure theory, both zero and infinity are meaningful measures. But you’re talking about philosophical concepts that I can’t comment on.
Please don’t say I said things that I did not say. ,999… = 1. Period. I said nothing else. And I qualified this by saying that this is a theorem in standard math. Nothing more. Nothing to do with the real world, nothing to do with philosophical notions of existence. I did not say “1 infinity.” I said .999… = 1, and this is true in the mathematical real numbers, which have nothing at all to do with the physical world.
I already told you that applying physical concepts like cm and mm is meaningless in this discussion. I can’t respond to you if you ignore this point. But to only one decimal place, .9cm and 1cm is certainly meaningful, at least to the limit of measurement precision. And that difference is .1cm. Not infinity. I have no idea what you’re talking about.
I’m afraid we’re not on the same page. Perhaps others can speak to you about Existence. I can’t. It’s all I can do around here to clarify some of the math and I have little enough luck with that.
That’s only true if the infinite series is absolutely convergent. If a series is only conditionally convergent – meaning that it converges, but the corresponding series made up of the absolute values of the terms doesn’t – then you can rearrange the terms of the series to attain absolutely any sum you desire! This is the amazing Riemann rearrangement theorem.
For example the alternating harmonic series 1 - 1/2 + 1/4 - 1/4 + 1/5 - 1/6 + 1/7 - … converges. In fact it converges to ln(2), the natural logarithm of 2.
But the corresponding series made up of the absolute value of the terms 1 + 1/2 + 1/3 + 1/4 + … diverges. (Freshman calculus again). So in fact you can rearrange the terms of 1 - 1/2 + 1/4 - 1/4 + 1/5 - 1/6 + 1/7 - … to make it sum to any value you like.
@obsrvr524 You and I agree on this! We’re two against @Magnus. How about that LOL!
First let me introduce myself a little here. I have been casually studying James’ posts at a distance for years, long before he started posting here. I have a very large, although incomplete library of his posts from many boards (probably around 30,000 posts). And as merely a hobby I have been sorting through watching specifically for errors in rationale or logic. I am a bit intrigued because of many of the things James’ spoke of from many years ago that people just don’t ever bring up - his “Affectance Ontology” giving a meaningful definition to what it means “to exist” being an obvious example - why light travels at that specific speed - why the universe exists at all - why positive and negative physically (and philosophically) attract - and many more.
James was what I/we call an “analytical reductionist” - someone who reduces topics to their fundamental essence and builds up from there. So logic is very important and as he pointed out himself, logic begins with definitions. So he defines a lot of things apparently trying to be as precise as possible and seeming to do a pretty good job. He admitted on several occasions to not knowing that some of what he was explaining had already been explained use different terms and there was no substantive difference other than the terms. On other issues he points out why his definitions are actually more revealing and clear - leading to deeper connective revelations.
So when it comes to maths, James reduced a few issues to the fundamental defined terms of concept and logic (noting that maths is just logic applied to quantities). And as a part of that he used hie “infA” term which he defined as -
infA = [1,2,3,4,…]
I noticed by continued tests that he really meant for infA to be the “size” of the natural number set, not the set itself. And he explained his reason for using that terminology rather than standard maths. And it seemed a pretty good reason. In a some newer posts on another board I found that he actually showed how usual maths’ hyperreal notation was actually exactly matching his own just with different letters and symbols.
So enough of that.
You are making a claim that I do not believe you can justify or logically prove. Specifically this -
“there is a bijection between the naturals and events, given by (f(n) = 2n)”
I think we can agree that “bijection” means a 1to1 correspondence, right?
Assuming so, I would like to see how you prove that your (f(n) = 2n) is applicable to an infinite set. What you are claiming is that -
(f(\infty) = 2\infty)
I don’t think you can justify or prove that is the case.
Your up. - Be sure to include the necessary well-defined terms - else - you know what.
I have to conclude from that you mean that there is an infinity of something between those measures. I can agree with that if that is what you mean. But I don’t see the relevance of it.
Or is it that you are trying to say that because there is something infinitely numerous between literally ALL things, Infinity (as a thing) is omnipresent and thus has something to do with God?
Thank you for this interesting post. I had no knowledge of James till I stumbled onto this forum and the .999… thread. I did gather that math was a very minor interest of his and that he was mostly interested in physics and ontology. In the process of conversating (ie arguing) with him about this subject I did spend some time reviewing some of his other threads, but not much. It’s fair for me to state my bias here. I regard James as a prolific crank. This is my opinion, not something it’s worth arguing about. I’ve always been an avid mathematical crankologist. The circle-squarers, the Cantor-deniers, and so forth. Not so much the physics and philosophy cranks. Cranks are generally smart people with a technical background, but they just have the crank gene. And often a lot of energy. They’re 100% male, I have never encountered a female crank. The circle-squarers and angle-trisectors are all retired engineers, men who had successful careers in disciplines involving numerical precision but who lacked the math gene somehow. To me, James was a classic case. That’s my observation, that’s my opinion, take it as you will. I am amazed that he left a trail of acolytes on this forum. To me it’s a bizarre phenomenon.
As I say no rebuttal is needed, I merely state my opinion.
In this thread (I interacted with him in no others) he continually said to me that something followed from “logic.” But when I tried to pin him down, “Do you mean classical logic, sentential logic, first-order predicate logic, some flavor of intuitionist logic, etc.?” he’d always ignore the question, then attack me by saying that his (incorrect IMO) statements followed from “logic.” So I don’t think James was interested in logic per se, the word was just a cudgel he used to beat people over the head with.
I really don’t mean to pile on, and he’s not here to respond back so anything I say is unfair in that regard. I can only give my impressions of my limited interactions with him several years ago in this thread; and beyond that, it’s better if I stick to responding to what you and @Magnus and others post. I truly have no idea what James thought about infA and infB, only that they didn’t make sense to me at the time and that he was definitely using them incorrectly with regard to the hyperreals. But going forward I can’t discuss what he might have thought, only what present participants say.
He was a crank, arguably insane. That does not mean he was not prolific, interesting, insightful, entertaining, and a worthy debate opponent. But if you insist on classifying him, you know where I stand. And again I only refer to his mathematics. I know nothing of his physics or philosophy. Maybe he made more sense in disciplines where I hadn’t read his work.
As I say he often said to me that something is true by “logic” but I could never get him to say what that meant. There’s paraconsistent logic, constructive logic, modal logic, so many flavors of logic.
Not from where I sit, at least not regarding mathematics.
Perhaps. But if infA = (\aleph_0) then infB makes no sense at all. And if infA is a hyperinteger in the nonstandard reals, James’s understanding was deeply shallow, if I may put it that way.
I will credit James with one thing, and that is motivating me to make a serious mathematical study of the hyperreals. Once I did, I had technical facts to back up my sense that he was misunderstanding them greatly. But (as @Magnus points out) I did not discuss those issues here. I’ll agree that I probably did not refute or clarify infA and infB to my or anyone else’s satisfaction back then.
I can’t comment on the quality of James’s work on other issues. As regards his remarks in this thread, like I say, I’m on the record. But he’s not here to defend himself so I shouldn’t say any more about him. Other than to thank him for getting me to buckle down and learn the technical details of the hyperreals.
Ok. I think we’re all agreed to that, with the provision that I don’t know what the square brackets are here. Is [1, 2, 3, …] a set? An ordered list? Or what exactly? In my response to @Magnus tonight I pointed out that (\aleph_0) is the set of natural numbers, also known by its more familiar symbol (\mathbb N = {0, 1, 2, 3, 4, …}) where the order relation is not implied. That is, any reordering of these elements is still the same set.
And (\omega) is the ordered set ((\mathbb N, \leq)), the natural numbers in their usual order. I hope nobody’s troubled by my including 0 in the natural numbers. That’s standard, but if someone doesn’t like it just throw 0 out, it makes … zero difference to the discussion.
If infA is one or the other of these, I’m fine with that. But if you mean something different by using the square brackets, just tell me.
And … setting up for reading ahead here … the symbol (\infty) is (1) not defined; and (2) definitely not a member of the natural numbers. This is important in what follows.
There’s a “logic” problem here. You say James said logic is about definitions. But what is the definition of size? I know what is the size of the collection of fingers on my hand or players on a basketball team (5 in both cases). But I don’t know what is meant by the “size” of an infinite set. It could be the cardinality, or it could be the natural density (I explained that in my post to @Magnus earlier) or it could be its counting measure, which in this case is (\infty). But in measure theory we DEFINE the symbol (\infty) so that the counting measure of the natural numbers is (\infty). But the symbols is NOT a natural number and it DOESN’T carry all the other baggage people are burdening it with. It’s a symbol carefully defined to have the formal properties we need, WITHOUT introducing contradictions.
So (sorry I’m being longwinded here) we DEFINE the size of the natural numbers as its cardinality, (\aleph_0). So when we say size in set theory, we implicitly mean (unless we say otherwise) the cardinality of a set, which has a very particular technical meaning. It’s WRONG to say infA is the size of a set as if that’s a discovery or fact. Rather, it’s a DEFINITION; one that I understand to be synonymous with cardinality. But I don’t know if that’s what James meant, or if that’s what you meant.
And when @Magnus tells me that infB is the “size” of the even natural numbers, then I know something’s wrong; because the even numbers and the natural numbers have the same cardinality.
If his explanation was logically consistent I’d be fine with it, but so far you have not gotten me to that point. You’ve said that infB is the “size” of the natural numbers, but you haven’t defined size for infinite sets. If I guess you mean cardinality, then infB is redundant. If you mean something else, you haven’t said what that something else is. And once someone says, “Oh well I am not going to do standard math,” then I am on the alert for either (a) crankery; or (b) something new I might learn about. So far I’m not seeing (b).
I’ll reserve judgment. But right here at this point, you have defined infA as the “size” of the set of natural numbers, but you have not defined what you mean by size. If if you mean cardinality, then it’s no different than infB. And if you mean something else, you haven’t told me what that is.
His references to the hyperreals in this thread (which is my only exposure to James’s work) were flat out wrong. If he later educated himself about the hyperreals, all to the good, but I have no knowledge of that one way or the other.
Ok. I feel slimy dumping on James, who isn’t here to defend himself and is a far more prolific and arguably smarter guy than me. I hope we can work with whatever you and @Magnus and others can say to me directly. We can’t consult James on his intentions.
Absolutely.
This is most disingenuous. But that’s a pejorative word and you clearly are writing in good faith so I shouldn’t say that. Let me just point out that:
The symbol (\infty) has not been defined; and
It is most definitely NOT repeat NOT a natural number.
The function (f : \mathbb N \to \mathbb N) is a function that
INPUTS a natural number; and
OUTPUTS a natural number.
So the symbol (\infty), whatever it might be, is most definitely not a valid input OR output to the function (f).
The function (f) inputs 0 and outputs 0; inputs 1 and outputs 2; inputs 2 and outputs 4; inputs 3 and outputs 6; and so on. Each input corresponds to exactly one unique output; and each output comes from exactly one input. That’s a one-to-one correspondence, or bijection. Thus (f) is a bijection between the set of natural numbers and the set of even numbers; therefore by definition, these two sets have the same cardinality.
I hope I’ve made my point. (\infty), whatever it is – and you have not defined it – is not a valid input to (f). The only valid inputs are the natural numbers 0, 1, 2, 3, 4, 5, …; and what comes out are 0, 2, 4, 6, 8, 10, …, respectively. That’s a bijection.
I want to add one thing. I don’t claim that the naturals and evens have the same “size” in any conventional sense. I mean ONLY that there exists a bijection between them. And then I DEFINE “same size” as having the same cardinality. So when a mathematician says two sets have the same size, they are making a technical statement, which SEEMS like they’re making some kind of philosophical claim. They aren’t. They only mean that the two sets have a bijection between them.
If it helps, it’s like when a doctor examines your spleen and tells you it’s “unremarkable.” That doesn’t mean you’re not a special snowflake, one of God’s own creatures. It means you don’t have some awful spleen disease. The word unremarkable means one thing in conversational English, and something entirely else in doctor-speak. Likewise size. In set theory it means cardinality, which means there’s a bijection. It doesn’t refer to conventional notions of size in everyday life, which is a great source of confusion I think.
_
Just a brief statement - James defined “logic” as “the consistency of thoughts and language” irrespective of categories (as I said - a serious philosophical reductionist).
Oh and he preferred “degree” of infinite to “size” of infinite.
I think I agree with every bit of that.
But there is the catch.
I hope we can agree that the natural number set is an infinite set. And that means that its items are infinitely numerous and that any representation of it must include that feature. So if you declare that (f(n)) represents each and every (n) within the infinite set, (n) must be allowed to become infinite.
If (n) is not allowed to become infinite then it does not represent each and every item in the infinite natural number set (consistency in representation must be maintained). And that directly leads to the issue at hand - what is represented in one set but not in the other.
I already regret my earlier intemperate remarks. He’s done far more for this board than I ever could and he’s not here to respond to anything I say. Going forward I’ll try to keep things in the present. Perhaps there are layers of understanding that I missed by only reading this thread and nothing else of his work.
Then “degree” requires definition. Is degree cardinality? Ordinality? Something else? Perhaps we can explore that going forward. What does it mean to you? Are you saying infA is the degree of the natural numbers? I’m perfectly ok with that. And that infB is the degree of the even natural numbers. I would just like someone to explain to me how they are not equal to each other, since they do have the same cardinality.
I have to assume that you’ve at least seen high school math, where they talked about functions. Perhaps they made you graph (f(x) = x^2) for (x) a real number; and they showed you that the graph is a parabola with vertex at the origin, opening in the upward direction. That’s a function applied to an infinite set, namely the set of real numbers. If we can’t agree that there are functions defined on infinite sets, we can’t make any progress at all. Is this something you’re disputing?
If I have a natural number (n), I can double it to obtain (2n). That’s because multiplication is a binary operation defined on the natural numbers. The product of two natural numbers is another natural number.
We can get to this doubling function in different ways. We could start with the Peano axioms for the natural numbers, and build up addition and then multiplication, and show that they have the familiar properties: addition and multiplication are commutative (a + b = b + a, ab = ba); multiplication distributes over addition(a (b + c) = ab + ac)) and so forth.
Or we can just agree to believe in the real numbers as presented in high school, with their addition and multiplication, and restrict our attention only to the nonnegative integers 0, 1, 2, 3, …
I truly hope we can take the positive integers and their addition and multiplication as given; as well as the fact that we can define functions on the natural numbers that output others. The doubling function, the squaring function, polynomials, the prime function (p(n)) that outputs the n-th prime, and so forth.
Are these things you deny, or whose existence you don’t believe in or don’t accept?
Do you or don’t you agree that given a natural number, I can multiply it by 2? And that this is true of any natural number 0, 1, 2, 3, 4, …?
I’m not making that claim at all. (\infty), whatever it is – and you have not defined it – is not a natural number. The natural numbers are 0, 1, 2, 3, 4, … Every single one of them is produced by starting with 0 and adding 1 a finite number of times. That’s the only way to make a natural number.
I’m claiming what I have to believe you already know and accept: That given a natural number, I can double it.
You are perhaps confusing properties and aspects of individuals, with properties and aspects of their collections or containers. I can ride a horse, but I can’t ride a barn. I can eat an apple, but I can’t eat an orchard. Surely this makes sense.
I can do things to individual natural numbers that I may or may not necessarily be able to do to the entire set of them. This seems like an obvious distinction to me. I can’t ride a barn and I don’t claim to be able to multiply the entire set of natural numbers by 2. Although to be accurate, we can define cardinal arithmetic on transfinite numbers. But that’s beyond us at the moment.
Yes. And by the way, how do we know that? What is your definition of an infinite set?
The standard definition (but not the only one) was proposed by Richard Dedekind in the 1880-somethings. He said that a set is infinite if it happens to be the case that it can be placed into bijection with a proper subset of itself. By this definition the natural numbers are infinite; since for example they can be placed into bijection with the even numbers. And the set {1, 2, 3, 4, 5} is not, because no matter how hard we try we can NOT place it into bijection with any proper subset of itself. A set that is infinite under this definition is called Dedekind infinite.
Well, I don’t know what you mean by saying that it’s infinitely numerous, but I’ll assume that you mean it’s infinite, say by Dedekind’s definition.
I don’t follow that at all. (\aleph_0), (\omega), (\mathbb N), infA, and {0, 1, 2, 3, …} are all representations of the natural numbers, and each of them is finite. In fact all human representations are finite. In this thread people have been throwing around the symbol (\infty) to represent the infinite, and although the infinite is big, endless, ineffable, and so forth, the symbol (\infty) is perfectly finite. It’s like an 8 lying on its side. All representations in human language are finite. So I don’t follow or agree with your point. Nor are barns anything like horses. I may have lost money betting on a nag at the racetrack, but nobody ever bet on a barn. Right? I am certain you must appreciate and agree with this point, that an aggregate need not and usually does not share the characteristics or aspects of the individuals it aggregates.
Each and every individual natural number is finite. It’s the entire collection of them that’s infinite. I think this is a point you are confused on, as witnessed by your attempting to explain infinite summation notation. All it means is that there’s an element for each of the natural numbers 0, 1, 2, … There is no “infinitieth” element.
Think about what you’re saying. Does 5 ever become infinite? No. Does 147 ever become infinite? No. Each natural number is finite. 1, 2, 3, 4, … You keep adding 1 to get the next number. In fact I hope you will click on the link I gave to the Peano axioms. They show how the natural numbers are constructed. There are two rules:
0 is a number.
If n is a number, the successor of n, denoted Sn, is a number.
Repeated application of these rules gives us an endless sequence of numbers: 0, S0, SS0, SSS0, SSSS0, etc. For convenience we invent some shorthand notations. S0 = 1, S1 = 2, S2 = 3, and so forth.
Each and every natural number is finite. Every horse has four legs and a tail. The collection of all the natural numbers is infinite. Barns have roofs but no legs or tails. Aggregates need not and generally do not share the attributes of the individuals that they aggregate.
No. Why do you say that? Barns don’t have tails. Horses have tails. Barns are collections of horses. But barns don’t have tails. This is a very obvious point.
Each natural number is finite. The collection of all the natural numbers is infinite.
The doubling function 2n multiples each natural number by 2, resulting in 2n.
You can’t ride a barn. The set of natural numbers is not a natural number. You carry your groceries home in a grocery bag (or sack, if you’re from the American south), but you eat the groceries and you don’t eat the bag. The bag is a container for the groceries, just like the set (\mathbb N) is a container for the natural numbers. Each number in the container is finite, and there are infinitely many of them. When you buy a carton of eggs, it contains 12 eggs. But there aren’t 12 containers. There’s one container. One container that contains 12 eggs. The set of natural numbers is a container that contains infinitely many natural numbers, each one of them finite. Aggregates aren’t anything like the individuals that they aggregate.
Let me see if I can circumvent some potential confusion here before we get lost in the bog (I hope).
Consider the following to be declared definitions for this discussion -
Degree of infinite refers to the amount of change from a chosen standard infinite set or sequence limit. If infA represents the chosen standard then infAset+1 (eg. {-1,0,1,2,3,…}) increases the degree of infinite from that standard.
(\infty) refers to the limit of an infinite succession. It is not within the sequence but the first successor that cannot be reached - an endless sequence plus 1 more. I think we can agree that such a number cannot exist. (\infty) is not a number but an approached conceptual limit.
So when I said that you have proposed that -
(f(\infty)) = (2\infty)
I was saying that you have equated their limits (by implying bijection and equal degree of infinite).
And the Piano axioms are not an issue with me.
Moving forward - I can’t allow the use of “Dedekind infinite sets” as a part of your proof because that would be presuming the consequent. A Dedekind infinite is defined as being the very kind thing I am asking you to prove exists. So we can’t first presume that it exists to prove that it exists.
I am concerned with your understanding of bijection.
I am far from being a mathematician so feel free to correct me but I understand that a bijection must work in both directions - each element in A must be able to correspond to each in B and also each element in B must correspond to each in A (else it would be either non-injective or non-surjective and not bijective).
If I understand bijection right, how are you going to show that the natural number set and the natural even number set are bidirectionally bijective? I don’t think that is possible (although the converse/negative seems tenable).
(f(n)|_0^\infty = (2n)|_0^\infty)
Of course merely stating that it is true isn’t proving that it is true.
Now that I see that, I agree. He did say that too. I am having to work mostly from memory and I remember once he stated it as the set and I had to figure out that he meant it as the specific infinite quantity. Your version is better. Thanks, mate.
I think the latter is true of me. Perhaps now I have a clearer understanding of what semantics you are focused on regarding this matter. Especially given the use of the following phrase: ‘degree of infinite’
Let’s suppose that that something, is atoms. It would seem you can have an infinity of atoms between those measures (just as you can have an infinity of measures). Dare I say the number of atoms could even change? Yes. You can have anything from 1 atom to an infinity of atoms. But can you have more than an infinity of atoms? When you say ‘degree of infinite’, is that something like solids and liquids in the sense that if the atoms between this measure were really compact and next to each other, there is more of them? Is that what you mean when you say degree of infinite? But isn’t the idea that there is more of them not contradictory?
The only reason you may be able to have an infinity of something between those measures, is because infinity is a measure or quantity that is true. If infinity is not a true measure or quantity, then you cannot have an infinity of any thing (measures, atoms, Existence etc.). Am I correct?
For there to be an infinity of atoms between those measures, something would have to encompass or accommodate those infinity of atoms. Given the semantics I think you are focused on, that thing would have to be a higher/purer degree of infinity. But this still doesn’t change the fact that that higher infinity would ultimately have be encompassed by Pure Existence (the truly Omnipresent. the purest Infinity). But here is my problem, which I would be grateful if you could solve for me:
Nothing can be ‘more Infinite’ than Infinity seems clearly true to me. Look back at the atoms example. Specifically at how there cannot be more atoms than an infinite number of atoms. If the atoms are so compact and together such that there is literally nothing between them, then what is the difference between atoms and infinitesimals or Existence? But Existence/Infinity does not have room for another Existence/Infinity. So there cannot be varying degrees of Infinity, can there? If yes, can you explain in atom form?
Also, suppose at time t, there isn’t an infinity of atoms between those measures. Given that it is impossible to count to infinity, how can it be possible to create or produce to infinity? How can an infinity of atoms exist between x and y, if they have not always existed between x and y?
In that post you have said several things I would have to object to. The first being that in reality you actually cannot have an infinity of atoms between any two locations. But for the sake of discussion and trying to gain common understanding, let’s assume that we really can get an infinity of gold atoms between location 0 and location 1.
And again, let’s give that degree of infinite a label - infA.
Now if there are infA atoms between locations 0 and 1, how many atoms are between locations 0 and 2?
Because we get bogged down with distracting semantic details, let’s do this one issue at a time like before. There are more issues to resolve afterward.
I agree that this is problematic. Which is why I was hoping that you would weigh in on this. This is the belief that I embraced to reconcile this problem:
You cannot have an infinity of any none-infinite thing between location 1 and location 2 (which we both agree on I think) including finite measures. But here’s where I think we differ:
At any given time, between location 1 and location 2, there is Existence (as opposed to non-existence). At time t1, there are no atoms between location 1 and 2. If God/Existence wanted to have an infinity of atoms between locations 1 and 2, there’d be no availableroom/time/measure to do this. Not only is it the case that something with a beginning cannot expand to the point of Infinity, but it also cannot increase in number to the quantity of Infinity. So we can never say there is an Infinity of something between locations 1 and 2 (not even measures). This is not the same as saying there is no Infinity between locations 1 and 2.
We do still have to say that there is Infinity between locations 1 and 2. Atoms could be added between locations 1 and 2 over and over again (forever even). They will still never reach Infinity in measure or quantity. Or in the amount of space they take, or in the quantity they are. Replace finite atoms with finite measures, and you will get the same result.
Since there are no ‘infinity of finite measures’ between any given locations, the problem is solved. There is always Infinity between any given location because Infinity encompasses all locations, measures (of which there is an endless number of), and separates one location from another. 1.9 cm and 2cm are separated by 1.95 cm. They are also separated by 1.955…cm 1.911…cm 1.999…m (and, this is the part where you’ll think me insane) 999…km. This is because when you suggest an infinity of digits, you suggest Infinity. You do not suggest just endless continuation of 9s being added to 999… It does not matter if it is m or cm or km. km… could denote Infinity if it logically implies the measure of Infinity. That you can have Infinity as a quantity, is an illusion. You have one Infinity. It is a measure. It encompasses all other measures. There is no end to the quantity of other measures. But this does not mean there is an infinity of finite measures.
I hope what I’m trying to convey is clear. My apologies if it is lacking or not concise enough. I would appreciate your feedback on the above.
I’m assuming this is for me. Lot of posts in this thread, helpful if you’d just quote me so I know it’s for me, thanks.
I’m sorry friend, this is just word salad. “Increases the degree of infinite from that standard?” Meaningless.
This is the definition of (\omega). Fine as far as it goes, but you are using it in ways it can’t be used. But essentially correct. However in the present context it’s irrelevant since as you agree, it’s not an element of the natural numbers hence not a valid input to the doubling function.
You just agreed that (\infty) is not an element of the natural numbers. So it’s not a valid input to the doubling function. Read what you yourself just wrote. Limits don’t make any difference here. The doubling function inputs a natural number and outputs an even natural number. It says nothing about things that are outside of the set of natural numbers.
No limits are involved. 2n is the plain old straight line y = 2x that you graphed in high school, a line through the origin with slope 2, restricted to the nonnegative integers. If you can’t grant me that, there’s no common ground to be had.
They show that every natural number is finite. There’s a sequence of them 0, 1, 2, 3, … Each one is finite. We can conceptually collect them into a set, or a class, or the extension of the predicate P(n) = “n is a natural number.” In so doing, we don’t add any numbers to the collection. We just conceptually collect a bunch of finite numbers.
Then what is your definition of an infinite set? How do you know one when you see one? Didn’t you say everything needs to be grounded in definitions? I proposed a definition of an infinite set. The idea I gave goes back to Galileo in 1538 and Arab mathematicians in the 1100’s. If you have a different definition I’m open to it.
I see. Should I laugh or cry?
I’ve been doing so, but you are not engaging with what I’m saying.
I have to take your remark as a sign of sincerity, but I do feel like Charlie Brown trying to kick the football as Lucy pulls it away. Is that reference still known to people?
Right. So in the 2n bijection, 1 corresponds to 2 and vice versa. 2 corresponds to 4 and vice versa. It’s clearly a bijection.
I’ve done so several times. 1-2, 2-4, 3-4, 4-8, … is a reversible bijection between the naturals and the evens. I can’t say it any more times. If you deny it, you need to show me a natural not paired with an even or an even not paired with a natural.
Meaningless notation. “Not even wrong” as the physicists say.
I can see by the smiley that you’re very proud of yourself. Or making a joke that perhaps I didn’t get.
It certainly wasn’t about pride. And I meant no offense, mate.
I did not say that it was an element of the set. I said that it was the conceptual limit of the set. A limit is never within the set. It is the first successor that cannot be reached so obviously is not within the set. And in this case isn’t even a number.
That would mean that the natural numbers can freely exceed infinite - else infinity is the limit (as proposed and well known).
I was certain that you understood (especially since dealing with natural numbers) that -
infinite ⇔ endless
There are many infinite sets that are not Dedekind-infinite so we certainly cannot use Dedekind-infinite as a definition for all infinite sets.
That is not a proof - far from it.
So as I suspected - you cannot prove it.
I could prove the negative but I can see that you are not interested. No worries.
