Question: taking one - the number - as a point on the number line, do you mean that there are more than one ones that are not one on the number line that are at a distance of zero from the point one - that are not one but some other number called one and understood to be one - or not one? Can you please make sense of this in terms a high-school mathematician can understand?The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case. — MoK
Um, no. Especially when you qualify it yourself as being hard for you to make sense of and very technical. Perhaps re-read my question - which I imagine you understand - and try answering in the same spirit and language.Perhaps this post help you. — MoK
You are correct in your observation. If the distance between two points is zero then we are dealing with the same point. This means that there cannot be a point between points since they are the same point. This is against the argument that there is always a point between two points on the real number. What I showed is that the distance between consecutive "means" (by "mean" I mean the point between two points) tends to zero. I hope things are clear now. If not please let me know so I would elaborate.That is, two points on the number line, zero distance between them: how can they not be the same point, or number, and if not the same, how can there be zero distance between them? — tim wood
Sure, gets smaller. Which in the very expression of which says that there is again a smaller - always. I suppose you can introduce a rule or limit that says at that limit the distance is zero, but then the point is identical with itself. I don't see how you get out of or around this.tends to zero. — MoK
I looked at all your posts and didn't find the proof that no non-zero real number is an infinitesimal. — MoK
The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every [non-negative] real number x there is a positive real number y such that y < |x|. — TonesInDeepFreeze
The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case. — MoK
I googled and I found two references about the division of cardinal numbers. You can find the references here.
— MoK
That source itself points out that when the numerator is less than the denominator, there is no definition of numerator/denominator. The very source you point to disputes your claim that (aleph_1)/(2^aleph_1) is properly defined. And you should have proven that for yourself when you first thought of it [here 'X' stands for the Cartesian product]:
K*L = card(K X L). And we have the theorem that if L <= K and K is infinite and L is non-zero, then K*L = K.
The definition of x/y:
x/y = the unique z such that z*y = x. If there is no such unique z, then x/y is not properly defined.
Suppose here that L <= K and K is infinite and L is non-zero:
If L = K, then there are Z such that Z*L = K, but there is no unique such Z. So L/K is not properly defined.
If L < K, then the unique Z such that Z*K = L is L. So L/K = L.
If K/L were properly defined, then K/L would be the unique Z such that Z*L = K. But there is no such Z at all, let alone a unique one. So K/L is not properly defined.
You did an Internet search but didn't even bother to read what you found, instead recommending that other people read it even though, unknown to you since you didn't even read it, it says the exact opposite of your claim that you made up out of thin air.
"(aleph_1)/(2^aleph_1)" is pure bunk. — TonesInDeepFreeze
This was a reply to the above comment from fishfry who claimed between any two distinct real numbers, there is always another one strictly between. The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case. — MoK
I am a retired physicist — MoK
It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. — TonesInDeepFreeze
I consider the number of the operation (by operation I mean dividing by two) to be strictly infinite. @TonesInDeepFreeze however claims that such an operation does not exist in the real number system: "There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0."Sure, gets smaller. Which in the very expression of which says that there is again a smaller - always. I suppose you can introduce a rule or limit that says at that limit the distance is zero, but then the point is identical with itself. I don't see how you get out of or around this. — tim wood
You didn't provide this argument before. Did you? You just defined infinitesimal!You need it spelled out for you again?
Definition: x is an infinitesimal if and only if, for every positive real number y, |x| < y.
Theorem: No non-negative real number is an infinitesimal.
Proof: Suppose x is a non-negative real number. Since x is a non-negative real number, |x|/2 is a positive real number and |x|/2| < |x|. So it is not the case that |x| < |x|/2|. So it is not the case that for every positive real number y, |x| < y. So x is not an infinitesimal. So no non-negative real number is an infinitesimal. — TonesInDeepFreeze
How could you have an infinite sequence of divided results without infinite division operations?You skip that I addressed that. There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0. — TonesInDeepFreeze
I was not confusing these. In fact, I mentioned the number of division operations to be strictly infinite.It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. — TonesInDeepFreeze
That is understandable but I was not arguing against that. I just argued that if the number of divisions is strictly infinite then you cannot get anything new by dividing the result further since the result is zero. I now know that the number of division operations cannot be infinite in the real number system. I don't know why!And the fact that the limit of the sequence is 0 does not refute that between any two distinct real numbers there is real number strictly between them. Indeed, the convergence to 0 itself depends on the fact that between any two distinct real numbers there is a real number strictly between them. — TonesInDeepFreeze
It seems that there is no operation of infinite division in the real number system. That was something I didn't know.This is incoherent at best, wrong at worst. I explained this to you at length. But look. You are trying to prove there are two points without a third between them, by claiming there are two points at a distance of zero. Can you see the circularity of your argument?
I explained this to you at length in a post you didn't bother to engage with. — fishfry
Oh yeah, I can guess that. We, physicists, work with the infinities all the time. Of course, mathematicians do not agree with how we deal with infinities but strangely physics works. :)I believe you. Physicists attempting to do math are often a source of humor and/or horror to mathematicians. But I'm sure you know that :-) — fishfry
I am not confusing the two.What Tones said. — fishfry
Yes, there is no infinitesimal in the real number system. I don't understand the rest of your comment.So if no real number is an infinitesimal, numbers are then what is relation to geometry. Is 2 then 2 points, or are all numbers a point? — Gregory
I didn't read Berkeley at all so I don't know what he is arguing about. Is he arguing that there is no infinitesimal in the real number system or he is arguing that there is no infinitesimal in any mathematical system? According to @TonesInDeepFreeze there are mathematical systems with infinitesimal.According to Wki both Cauchy (in Cours d'Analyse) and Edwars Nelson also compared infinite points to the numberline. Long before hyperreals i believe. The great writer and philosopher George Berkeley rejected infinitesimals on both mathematical and philosophical grounds — Gregory
I don't understand what you are trying to argue here.What about imaginary numbers, however? Stephen Hawking, in his attempt at find the wave function of the universe, proposed his (yep) No Boundary Proposal in 1983. I like to apply this "theorem" to consciousness. Hawking uses imaginary numbers to describe time as it goes backwards, behind the Big Bang. How are we to understand mathematically a state not having any boundaries? There is always a "here" and "there" in our experience. That is, except in consciousness wherein we can go deeper and deeper and we find no edge. The "limit" seems to be death, but in our experience we are infinite. Hence we can think about infinities.. — Gregory
The Kalam cosmological argument states that there cannot be an infinite number of past events therefore there is a beginning. The rest of the argument is about proof of the existence of God which I don't agree with.The kalam cosmological argument gives a great example of infinities embedded in another. The argument fails in its purpose because eternity, an infinity, contains all steps of infinity. There can be that infinity if there is the eternity. QED? — Gregory
The top of the cake looks like the top of the cake no matter how many times you divide the cake horizontally.If i cut a cake horizontally starting from the halfway point upwards with each slice being half the size of the one immediately below, what would the top of the cake look like? — Gregory
It is not.Isn't it indefinite? — Gregory
I don't understand what you are talking about.But you can definitely look at the cake, from all angles, and see that it has definite position in relation to its parts. So how do we reconcile the indefinite with the definite? — Gregory
We are not talking about space and time here. Whether space and time are continuous or not is the subject of other threads.I think this is what must be asked about the continuum. Hawking would say that four dimensional Euclidean space, with a time dimension that both 1) acts as space, and 2) is described by imaginary numbers, gives an answer to this question. That is to say, the universe as a whole gives the answer to the continuum. But how do imaginary numbers relate to geometry? — Gregory
Ok, I try my best to answer your questions. Perhaps, others (mathematicians @TonesInDeepFreeze, @fishery, and @jgill) would participate and answer your questions in a simple manner.I have already qualified myself as a high-school "mathematician" - that being why I try to make sense in English. — tim wood
No, I am not confusing the ideas of the number and the limit.Let's try this: it seems to me you are confusing ideas of number with limit. — tim wood
There is no last digit. The square root of two is an irrational number. The Irrational set, the set of all irrational numbers, is a subset of the real number set. Almost most of the real numbers do not have the last digit (the number of digits is infinite).For example, the decimal expansion of the square root of two goes on forever, is infinite, so what is the last digit? — tim wood
Bigger than any countable number.You want to divide something a "strictly" infinite number of times: great, how many times is that? — tim wood
I don't understand your objection to my comments. I used "strictly" when I wanted to see what is the result of an interval divided infinite number of times by two. Mathematicians think that there is no operation of infinite division in the real number system. I don't know why and I asked @TonesInDeepFreeze for an explanation. Perhaps he can answer my question in a simple term so you can understand as well.Just so, so when you talk about something that "strictly" can be, but which cannot itself be, then you tell me what the sense is. — tim wood
That's what I understand as a limit (process) - that can be approached but never reached, but if it could be reached, would yield whatever - the process of whatevering being nothing in itself, but useful as a tool.what is the result of an interval divided infinite number of times — MoK
How could you have an infinite sequence of divided results without infinite division!? — MoK
You didn't provide this argument before. Did you? You just defined infinitesimal! — MoK
One more time: No non-zero real number is an infinitesimal. The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every real number x there is a positive real number y such that y < |x|. We don't need to keep going over this over and over. — TonesInDeepFreeze
According to TonesInDeepFreeze there are mathematical systems with infinitesimal. — MoK
Bigger than any countable number. — MoK
Better, read Enderton's beautifully written 'A Mathematical Introduction To Logic' — TonesInDeepFreeze
Is f(infinity) a member of the above sequence? If yes, what is its value? If not, how could the sequence be an infinite one?So we have division, which is a binary operation. But we also prove that for every positive real number r, there exists a function f whose domain is the set of positive natural numbers and such that, for every positive natural number n:
f(1) = r
f(n+1) = f(n)/2
f is a function from the set of positive natural numbers into the set of real numbers. — TonesInDeepFreeze
Let's wait for your answer to the previous questions.Notice that it is trivial to prove that for no n is it the case that f(n) = 0. — TonesInDeepFreeze
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