Hi wtf,
You might be onto something here. I am thinking about Von Neumann and Godel numbering, but I need to give it some more thought.
However, with this specific example aren’t you getting into some trouble representing an uncountable object with a countable object? In this specific case I think you need sequences with limits.
Ed
Hi wtf,
I think I have screwed up my comments abount countable/uncountable. Though I am still not certain about your representation for Pi - 3.
Ed
Perhaps some confusion can be cleared by signaling to the matter, or it’s corresponding idea relating to languages in general, mathematics being a quantifier, leading to this proposition:
'formula beginning with a quantifier is called a quantified formula. A formal quantifier requires a variable, which is said to be bound by it, and a subformula specifying a property of that variable.
Formal quantifiers have been generalized beginning with the work of Mostowski and Lindström.’
As far as generalization is concerned, it appears to pair with a tendency to integrate sets that functionally demand such, to qualify within reasonable set specifications.
So the function may be set differently,
but it tends to integrate within some mixed set consisting of both: of specified and more general characteristics.
At least this is what appears to be implied here.
I may be way off with this generalization, it seems credible.
Hello.
I take (0.999\dotso) to represent the same thing as every other decimal number which is a sum of the following form:
(\cdots + d_2 \times 10^2 + d_1 \times 10^1 + d_0 \times 10^0 + d_{-1} \times 10^{-1} + d_{-2} \times 10^{-2} + \cdots)
Every (d_n) represents a decimal digit which is an integer from (0) to (9).
“Function” and “number” do not normally mean the same thing. You can make them mean the same thing, of course, but then, you must not equivocate.
How much money do you have? I have (f(x) = x^2) money. What does that mean? Normally, it means nothing. But of course, if you have a need to, you can make it mean something.
You can use horses to represent numbers. You can say “This kind of horse represents this kind of number”. For example, you can say that pegasuses represent number (1,000), centaurs number (100) and ponies number (10). This allows you to do arithmetic with horses. You can say “A pony multiplied by a centaur equals a pegasus” without being wrong.
You can do the opposite too. You can say “This kind of number represents this kind of horse”. You can say number (1,000) represents pegasuses, and because pegasuses can fly, you can conclude, without making a mistake, that (1,000) can fly too.
It’s all fun and games until you equivocate.
For example:
Horses qua numbers are indeed shapeless, but what is argued here is that horses qua animals are shapeless, which is not true.
In the same way, (0.999\dotso) qua limit is indeed (1) but what is being argued is that (0.999\dotso) qua sum is (1), and that is not true.
You’re not? What do you think a decimal expression like .1415926… means? It’s a map from the positive integers to the decimal digits. 1 goes to 1. 2 goes to 4. 3 goes to 1.
The expression is then mapped to a convergent infinite series by summing 1/10 + 4/100 + …
I cannot imagine you not knowing this. Please explain where you’re coming from. You have me totally confused.
Then I’m confused. If the question is, “Is .999… = 1 true in standard math?” the answer is yes without a shred of doubt. I could point you to a hundred books on calculus and real analysis. If we’re talking about standard math, how could anyone hold a different opinion?
But yes! In which case .999… = 1 in standard math and there is no question or dispute, other than to clarify for people what the notation means and why it’s true within standard math. It’s a theorem in ZF set theory. It’s a convergent geometric series in freshman calculus. It’s even true in nonstandard analysis, which some people aren’t aware of. There’s just no question about the matter.
So you really have me puzzled, Magnus. If you agree we’re talking about standard math, what is the basis of your disagreement?
Ok. So you admit that you are NOT talking about standard math, but rather about your private nonstandard use of mathematical symbols. In which case you can define .999… = 47 and I would have no objection. If that’s one of the rules in your game, I am fine with it; just as I learned to accept that the knight can hop over other pieces in standard chess.
You have already said that you are talking about standard math AND that you are talking about your own private nonstandard math. It’s not hard to misunderstand you!
Now let me talk delicately about infA. When I came to this forum several years ago, James was already a prolific poster, an ILP Legend to beat all ILP legends. I am reluctant to criticize him since he is not here to defend himself. He has far more mindshare on this forum than I do. I respect his prolific output, if not always its content.
That said, the concept if infA is confused and wrong in the extreme. The idea seems to be some sort of mishmash of the ordinal numbers, in which we do “continue counting” after all the natural numbers are exhausted; and nonstandard analysis, in which there are true infinite and infinitesimal numbers. The infA concept borrows misunderstood elements from each of these ideas and simply makes a mess.
One really valuable thing I got from this thread a few years ago is that James caused me to go deeply into nonstandard analysis, to the point where I understand its technical aspects. For that I appreciate James. But the infA concept is just bullpucky, I don’t know what else to say.
It means exactly what I’ve described to @Ed3. It’s a particular map from the positive integers to the set of decimal digits; a constant map, in fact, in which f(n) = ‘9’ for all inputs n. We then interpret this symbol as a real number as in the theory of geometric series, in which it’s proved rigorously that .999… = 1.
Again I agree that if you choose to make up a new system in which .999… has some other meaning, you are perfect within your rights. After all there do happen to be many variants of chess; played on infinite boards, or with a new piece called the Archbishop, etc. If someone enjoys playing alternate versions of standard games it’s ok by me.
Yes ok. Then we are done!
I fail to follow that. Did you learn geometric series at one point? The definition of a limit? I can’t tell where you’re coming from.
Actually the limit is defined as the sum. That’s what a limit is. It’s a clever finessing of the idea of “the point at the end” or whatever. You are adding your own faulty intuition. If you would consult a book on real analysis you would find that the limit of a geometric series is defined as the limit of the sequence of partial sums; and that the limit of a sequence is defined as a number that the sequence gets arbitrarily close to. I explained all this to @Phyllo the other day. That’s the textbook definition. You’re just wrong in your impression, either because you had a bad calculus class (as most students do) or none at all. It’s not till Real Analysis, a class taken primarily by math majors, that one sees the formal definition and comes to understand that the sum IS defined as the limit of the sequence of partial sums. That cleverly avoids the confusion you’ve confused yourself with.
Ah, the evil cabal of mathematicians. I will fully agree with you that most math TEACHERS form an evil cabal. One doesn’t get all this stuff sorted out properly till one sees the formal definitions; at which point, one learns that the limit of a sequence is defined by arbitrary closeness. The belief you have is a bad intuition that mathematical training is designed to clarify. It’s sad that we don’t show this to people unless they’re math majors, and you can rest assured that when I am in charge of the public school math curriculum the teaching of the real numbers will be a lot better.
Till then, I apologize on behalf of the math community that you weren’t taught better. But limits are very rigorously defined and your idea is just wrong.
Again I hope I’m not coming on too strong, I’m criticizing your ideas and not you. I know you are sincere. Except for the part where you say you’re talking about standard math AND that you’re not. That point confused me.
As I say, due to JSS’s extreme prolificness (if I may coin a word) on this site, plus the fact that he’s not here to defend his ideas, I prefer not to argue with him due to basic fairness.
So if you could frame your points from the beginning and not ascribe them to JSS, then I won’t be in a position to try to understand the ideas of JSS, which I found faulty four years ago. Just explain your ideas to me in your own terms. Else I’m arguing with someone who can’t argue back.
Again, as I’ve said many times, if you would consult a book calculus or real analysis, you would know that .999… represents the geometric series 9/10 + 9/100 + … whose sum, as defined in real analysis, is 1. I know you have an infinite series on one hand and a number on the other, but they are indeed equal mathematical objects, and this can be rigorously proved from first principles.
I do desire to understand the nature of your disagreement. But as you keep falling back on claiming that the standard definition of a limit is a lie, without providing any more supporting details, I remain puzzled. The definition of a limit is what it is, as is the way the knight moves. Rules in a formal game. They can’t be right or wrong, they’re just formal rules that have turned out to be interesting and (in the case of math) useful in understanding the world.
That can not be rationally disputed. One need only consult hundreds if not thousands of math texts that describe the standard definition of limits.
I could see your saying that mathematicians have gotten it wrong. But I can’t see your saying that they didn’t say what they did say!
It is not necessary for a sequence or sum to “attain” its limit. That’s the whole point of limits. Attainment is NOT part of the definition. Arbitrary closeness is.
Likewise, SOME properties are preserved when we pass to the limit, and others aren’t. For example in the sequence 1/2, 1/3, 1/4, 1/5 1/6, … each term is strictly greater than 0, But the limit is 0. It is a fact taught to math majors that when we pass to the limit, (<) becomes (\leq) and (>) becomes (\geq).
The terms of 1/2, 1/3, 1/4, … get arbitrarily close and STAY arbitrarily close to 0. That is the definition of the limit. You have faulty ideas because you haven’t grokked the formal definition of a limit. That’s all I can see of your objection.
That’s just something you made up. It’s not mathematically true. Limits are based on arbitrary closeness, not attainment. It’s perfectly clear that 1/2, 1/3, … never “attains” the value 0. The limit is 0 because the terms get arbitrarily close to 0. That is the definition. And even though each element is strictly greater than 0, the limit is 0. That’s how limits work.
It’s my own psychology that I question! It’s a lot like flat eathers. I think flat earthers are harmless and usually trolling. I think “scientifically minded” people who expend energy trying to debunk flat earth theory are sillier than the flat earthers. The flat earthers are having fun and the debunkers are way too serious for their own good, like Neil deGrasse Tyson, who got into a lengthy public dispute with some rapper about flat earth theory.
Yet here I am doing exactly the same thing. If someone disbelieves .999… = 1 it’s harmless, for one thing, and I’ll never convince them otherwise, for another. Yet here I am. I’m nuttier than the .999… deniers. I admit it
But thanks for agreeing with me!
I have three more examples I forgot to mention earlier.
Number are often interpreted as operators. If we interpret the number 5 as an operator that stretches a line segment, then the number i is an operator that rotates a line segment in the plane. That is in fact the best way to view it. So it’s common fro numbers to be identified with functions.
We have function spaces such as the set of all continuous functions that can be added and multiplied pointwise. With these operations of plus and times, the continuous real-valued functions of a real variable become a commutative ring.
Likewise we have famous function spaces like Banach and Hilbert spaces, in which functions are points, we have inner products, and we can do linear algebra on them.
I didn’t represent an uncountable object with a countable one. There are as many continuous functions (reals to reals) as there are reals. Easy proof. That was my only point.
I’m confused about where you’re coming from. What point are you making? Morally, functions and numbers are different. But in practice we use functions as numbers and numbers as functions all the time.
No.
I disagree.
The point of contention is the standard meaning of the symbol that is (0.\dot9).
Obviously, I have to repeat it at least one more time: I am NOT talking about what I mean by (0.\dot9), I am talking about what mathematical establisment means by (0.\dot9).
That said, you might want to argue that I am wrong in my belief that mathematicians define (0.\dot9) as a sum and not as a limit.
The limit of a sum is not the sum itself.
You can call the limit of a sum by the name “sum” – and people already do that, I understand – but that doesn’t erase the difference between the concept of a sum and the concept of a limit.
Let me give you an analogy. You can call numbers horses. There’s nothing wrong with that. But you can’t say there’s no difference between numbers and horses.
The argument I’m putting forward is that (0.\dot9) represents THE RESULT OF A SUM and not THE LIMIT OF A SUM. They are two related but different things.
That said, you might want to argue that mathematicians interpret (0.\dot9) as the limit of a sum (and not as the sum itself.) By doing so, however, you would be making an exception for (0.\dot9) and similar expressions because all other decimal numbers are normally interpreted as sums.
The result of a sum is a number that is attained. The limit of a sum is a number that is approached – not necessarily attained.
Your best bet is to argue that (0.\dot9) represents a limit rather than a sum. You won’t get far by trying to deny the fact that the concept of limit and the concept of sum are two different concepts.
en.wikipedia.org/wiki/Series_(mathematics
You haven’t added anything other than reiterate that you are unfamiliar with the standard mathematical definition of the limit of a convergent series.
FWIW I’ll outline the logic.
(1) First we define the limit of a sequence a1, a2, a3, … by the “arbitrarily close” standard, which is formalized by the business about the epsilons that you may have seen. (I’ll skip writing the gory details unless requested).
(2) Then we would like to define what we mean by a notation like a1 + a2 + a3 + … The axioms for the real numbers only allow us to add up finitely many real numbers. So we have to DEFINE what an infinite sum is.
We form the sequence of partial sums: a1, a1 + a2, a 1 + a2 + a3, + … Each term is well-defined because it’s a finite sum of real numbers. Now if the resulting SEQUENCE of partial sums has a limit as given by (1), then we define the infinite sum as the limit of that sequence. It’s a definition.
That’s it. That’s really it.
Now if the question is whether that’s what mathematicians say, it is. Wikipedia agrees and so do hundreds of textbooks.
If the question is whether the mathematicians maybe got it philosophically wrong, that’s a different discussion; and one that I’m not entirely unsympathetic to. But you claim to dispute that this is how mathematicians define infinite sums, and you’re just wrong about that.
JSS started this thread. It’s his topic. That’s why I’m mentioning him. You are wondering what this thread is about. Since it’s James’s thread, it’s James who decides (or rather, who decided long time ago) what this thread is about. And what this thread is about is standard math.
I am not saying that the definition of a limit is a lie. I am not sure where you got that from. What I’m saying is that mathematicians claim that (0.\dot9) qua sum is equal to (1) and that they do so by a variety of means one of them being by pretending that an infinite sum and the limit of an infinite sum are one and the same thing. Basically, what I’m saying is that they are equivocating (not merely working with different definitions.)
I am not talking about limits, I am talking about sums. Sums aren’t based on arbitrary closeness.
I don’t think you got that right.
The point of contention is how mathematicians define (0.\dot9). (Not how they define the term “infinite sum”.)
Your claim is that (0.\dot9) represents the limit of the sum (which you also call “the infinite sum” or “the sum”.)
My claim is that (0.\dot9) represents the infinite sum itself and not the limit of the sum.
Does that help?
James’s idea of infA is not only not standard, it’s logically incoherent. As were his other ideas.
I gave you the Wiki link and the outline of how the definition works. It’s logically solid.
But first you say that you are merely disputing that mathematicians define infinite sums the way I say they do.
But now you say they are equivocating, meaning that you disagree with the official definition. So you ARE challenging the official standard definition.
I just gave you the Wiki link. Sequences are based on arbitrary closeness; and infinite sums are defined as limits of SEQUENCES of partial sums.
It’s described with perfect clarity on the Wiki page.
“A series ∑an is said to converge or to be convergent when the sequence (sk) of partial sums has a finite limit. If the limit of sk is infinite or does not exist, the series is said to diverge.”
en.wikipedia.org/wiki/Series_(mathematics#Convergent_series
The infinite sum IS the limit of the sequence of partial sums. Read the Wiki page if nothing else.
How can you read that page and claim it says something other than what it says?
The fact that James’s (infA) is not a standard mathematical representation does not mean this thread has nothing to do with standard math.
I already told you I am not talking about limits. What’s the problem? Note that when you say “infinite sum” you mean “limit”.
When you say “infinite sum” you mean “limit”. When I say “infinite sum” I don’t mean limit. I mean what the words the term consists of suggests: a sum consisting of an infinite number of terms. That’s precisely how (0.\dot9) is interpreted by mathematicians: as a sum consisting of an infinite number of terms. That’s what I claim. You get distracted by the fact the word “infinite sum” has been given additional meaning – that of a limit.
It has no meaning except by virtue of how it’s defined. How do YOU define “a sum consisting of an infinite number of terms?” How do you define it? Lay out your formal definition and we can kick it around. I’m pretty openminded. But you keep saying there’s some secret definition that you won’t share with me.
Bear in mind that the axioms of the real numbers only provide for the addition of finitely many terms. To define an infinite sum we must do exactly that: define it.
How do you define it?