When Euclid said a point was that without length or breadth, and that the intersection of two lines was a point, are these facts about the world in your opinion? It seems to me they are manifestly not.
They are rather facts about abstract entities (lines and points, circles and triangles) that do NOT exist in the world. Is it “true” in any meaningful way that the angles of a triangle add up to 180 degrees?
Why must .999… have some referent in the world? There are no circles or triangles or dimensionless points or breadthless lines in the world, right? Perhaps I am misunderstanding you in some way.
We are literally making loops, circles from inductive argument to into deductive type. The point is uber cardinal identity, and that is the point which Frege tried to show logically, the identity of infinities has been transcribed to a comparison by infinitesimally similar eigenvalues.
There is no literal language to describe it.
There is a profound but very subtle albeit unnoticeable difference here. It is extremely difficult to get to this distinction because it is at the very heart of the argument where the argument folds back on itself, via the reduction ad absurdism implicit within.
The proof is more or less in the pudding, and it becomes obvious that here the basic differentiation between philosophy as an empirical specification of values is indifferentiable on a basic logical field.
As far as the slights of mathematical preparation is concerned, this should dampen the enthusiasm for pointing to it, because, of this correlation, but I take some comfort in the thought, that this is basically a humor laden attempt to set a vector toward the ultimate aim of this forum: to keep a dialogue going, and not to come to an earthshaking final reproach, when we may guess the real reasons we are on this forum, where to some of us, this may be new (like me), while to some, it serves as a kind of refresher.
I’m very familiar with that paper. The title is misleading. The inequality he proves is a technical construction that differs considerably from the usual meaning of (999…) In effect he demonstrates the hyperreal analog of the familiar fact that, for example, (.999 < 1).
Then he changes notation to hide some conceptual difficulties and presents the result as a better way to teach. He is making a pedagogical point. The paper does not shed light on the mathematical or philosophical issues. (.999… = 1) is still true after he’s done.
Professor Katz is an authority on infinitesimals. In this case he wrote a very interesting and unobjectionable paper, but he let his enthusiasm/advocacy get the better of him, and he wrote a misleading title. I am unhappy that so many people see the title on Wiki and are misled.
If anyone’s interested I can talk a lot more about this. The real value of the paper is that it provides an excellent introduction to the hyperreals. On that basis I recommend reading it. Forget the title. It’s really stretching the point.
We are not at all talking about math here. We are talking about why you believe something (that was the question that I asked). Or are you saying that you believe that infinite sets have the exact same size ONLY because you read it in a book? That would certainly be true of wtf.
And if I say that there are two circles, each of diameter 1", are you going to tell me that in fact there is only one circle because they are identical? Only one in the entire universe? I proposed that there are two sets that happen to be identical. But no, being totally disingenuous, you have to proclaim that you can’t comprehend that and that there can only be one set.
You are obviously being 100% disingenuous (as is your usual - “deny everything at all cost”).
You don’t have a ‘proof’ which is applicable to general sets. You’re showing something about identical sets - two identical sets have the same elements and the same size. Seems to be an obvious property which requires no demonstration.
Our disagreement here appears to be semantic. Whether there are two identical sets or two labels for the same set, my answer remains the same: if you take two sets that are identical, and add identical elements to each, the sets will remain identical. Since the identity function is always a bijection between two identical sets, yes, there is a bijection between the two sets.
I largely agree, but Carleas is going to extremes to divert away from any future point to it all to which he might have to agree concerning infinity.
That is not in question. The proposal is that there are two. End of story.
The point to only that tiny proposed thought is that the notion that the two sets are still identical when extended infinitely is speculative induction. There is nothing wrong with that thought. No need to fear it or fear agreeing to it (since its pretty damn obvious).
… when infinitely extrapolated.
Now when I add another element to ONE of the TWO sets, an “A”, they are still infinite; P = {A,1,2,3…} N = (1,2,3…}
But now are they still identical?
Does the bijection notion still apply, disqualifying their equality in size?
Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.
And then that becomes entangled [for some of us] in how we think about mathematics in relationship to the “real world”. In other words, the existential interactions of human beings from day to day.
For many of the truly unsophisticated [like me] the part where mathematics ends and, among other things, philosophy, the natural sciences, religion, values, etc. begin, is, among other things, truly ineffable.
I follow the exchanges here and at times the arguments become quite fierce. So I can’t help but wonder 1] why there is not “a way” in which to finally resolve it and 2] what this all might possibly have to do with the things that interest me at ILP.
They are no longer identical, though they are still the same size, and there is still a bijection (f(x) ,, P \to N):
(f(x)=\begin{cases}1&x=A\x+1& x \neq A\end{cases})
wtf, I have a question that I think James raised earlier that I wasn’t sure how to answer and google’s no help, maybe you have seen it before:
Is there a different limit as (x \to ) different infinities? Is ( \infty ) in the case of a limit assumed to be ( \aleph_0 ), or does it not matter, or is the question meaningless in this context? I can’t see how it would matter, but it does seem unusual to say that it doesn’t matter.
I think James framed the question as something like, What’s the difference between ( \lim_{x\to\aleph_0} \frac{1}{x}) and ( \lim_{x\to\aleph_1} \frac{1}{x})? It’s a good question.
Caeas, I know that the question was not asked of me, but may I interject a Google article that suggests that Cantor’ s Paradox may touch on the topic , perhaps even help with deriving an answer.
Might as well go for it. I no longer respond to posts by or referencing the biggest troll on this forum.
Carleas I simply cannot imagine what you were thinking. Maybe you haven’t noticed the behavior of that individual toward me. Why would you mention his name to me twice? To see how much abuse I’m willing to take?
I’ve already drawn the line. Nothing personal to Carleas but I’ve had enough of James. And if you look back at his threads since 2010 you’ll see I’m not the first.
A bijection is a one-to-one correspondence. It is obvious that they do NOT have a one-to-one correspondence. Set N has a representative for all of the natural numbers. And those have a one-to-one correspondence with the same natural number elements in set P. But set P also has a letter “A” element, for which the set N has no corresponding letter.
Obviously they do not have a one-to-one correspondence. So how can they possibly be bijection true?
Of course! I asked wtf because he seems well-versed, but an answer from anyone better-versed than me (which is just about anyone) would be appreciated.
A broken clock is right twice a day. And so is an abusive clock.
For what it’s worth, James gets more abusive in response to better arguments, so while I understand the decisions to avoid discussions with him, you should take his abuse as a vindication.
Under (f(x)), A corresponds to 1.
Then show one element in either set that not does not have one and only one corresponding element in the other set under (f(x)).
What is “Under (f(x))” supposed to mean, besides merely another obfuscation? There is no (f(x) ) involved.
We had every element paired. We added to one of the two sets and not to the other. By definition of “adding to”, one set is now necessarily different than the other (else we did not add anything).
So far, there are two arguments against bijection of the sets P and N;
They were obviously paired with the natural numbers and then we added a letter to one of them
The very definition of “adding to” requires that one of them be different.
No, it isn’t a “strong argument”. It is a “hand waving” distraction.
So in an attempt to come up with a rationale for bijection, you are going to define an (f1(x) ) such that the first of your chosen function’s output is an “A” and then from 2 onward the output is the natural numbers, while (f2(x) ) is merely the natural numbers. So the only difference is that you are going to call them “functions”.
P = (f1(x) ) = A,1,2,3… N = (f2(x) ) = 1,2,3…
So even with your effort to rewrite the more obvious into a conflated function comparison, what makes you think that those functions are equal size? How do you know whether they have a bijection throughout infinity? Let me guess … You believe it because a book told you that if f(x) is infinite then it has bijection with any other infinite f(x), right? - assuming the consequent - circular argument - “it’s true because it’s true” - “it’s true because that is what they said”.
With my way, it is obvious that there is no bijection. With your way, you have to speculate what might be happening throughout infinity. And of course, in the face of the uncertainty that you injected, you choose to believe that there is a bijection so as to justify your position.