Hang on a sec, you're not trying to avoid those old record long threads of yours getting beaten? :) — jorndoe
In my book 0.9 + 0.1 = 1 and 1 - 0.1 = 0.9 and so 0.9 does not equal 1. There's a similar argument for 0.99 and 1 and so on. So far each element of 0.99999....., I have an argument that it does not equal 1. However, I see that your proof involves limits and I know that in that context words change their meanings. So I'm curious. — Ludwig V
3×13=1 and 3×0.333...=0.999... — Michael
That makes 0.999999..... = 1 just an illusion created by the notation you have decided to use. It is not a proof. In my opinion. You might have a different idea of what a proof is. — Ludwig V
The argument that 0.999... only approximates 1 has grounding in formal mathematics. In the 1960's, a mathematician, Abraham Robinson, developed nonstandard analysis (Keisler, 1976). In contrast to standard analysis, which is what we normally teach in K-16 classrooms, nonstandard analysis posits the existence of infinitely small numbers (infinitesimals) and has no need for limits. In fact, until Balzano formalized the concept of limits, computing derivatives relied on the use of infinitesimals and related objects that Newton called "fluxions" (Burton, 2007). These initially shaky foundations for Calculus prompted the following whimsical remark from fellow Englishman, Bishop George Berkeley: "And what are these fluxions? ... May we not call them ghosts of departed quantities?" (p. 525). Robinson's work provided a solid foundation for infinitesimals that Newton lacked, by extending the field of real numbers to include an uncountably infinite collection of infinitesimals (Keisler, 1976). This foundation (nonstandard analysis) requires that we treat infinite numbers like real numbers that can be added and multiplied. Nonstandard analysis provides a sound basis for treating infinitesimals like real numbers and for rejecting equality of 0.999... and 1 (Katz & Katz, 2010). However, we will see that it also contradicts accepted concepts, such as the Archimedean property — https://files.eric.ed.gov/fulltext/EJ961516.pdf
I can see that point. I didn't look at the issue in the light of infinite series or take on board that it was a question of the sum of an infinite series. I apologize for the distraction.Then again, one could reject that the equation for the sum applies. The equation of the infinite sum relies on the notion of limit, and it is the notion of limit that is at play on the 0.999... debate. — Lionino
That's very neat.There is
a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1 — Lionino
Let's not distract from supertasks by questioning very simple mathematical facts. — Michael
It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine.Oh I see what happened. Ludwig brought up the old .999... = 1 chestnut in the staircase thread, and it apparently got moved over here to revivify this four year old thread. — fishfry
Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question.What number can possibly get between ALL the terms of that sequence, and the number 1? — fishfry
I look forward to mankind's return to the Garden of Eden.Now it is also true that 4/0=∞ and 9.7181=∞. And with a little more leg work I shall demonstrate that all numbers are actually equal to each other. Multiplicity is mere illusion, a result of the Fall and Adam's sin. — Count Timothy von Icarus
It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine. — Ludwig V
Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question. — Ludwig V
However, it is also true that 1 is the sum of the infinite series 0.999... - and therefore the limit. — Ludwig V
But an infinite series never reaches its limit. — Ludwig V
To put it another way, "=" in this context (an infinite series) does not mean what it usually means. — Ludwig V
Yes. I assume you mean all the terms of the infinite sequence?There is an answer. The answer is that there is no number greater than all the terms of the sequence, and less than 1. — fishfry
And I'm puzzled why you think I'm disagreeing with you.I think I'm a little bit puzzled that you have this confusion after I've explained it in the other thread. — fishfry
So it is. But what is the element of the sequence immediately preceding 1?The limit is equal to 1, in exactly the same sense that 1 + 1 equals 2. — fishfry
Yes. I assume you mean all the terms of the infinite sequence? — Ludwig V
And I'm puzzled why you think I'm disagreeing with you. — Ludwig V
[/quote]So it is. But what is the element of the sequence immediately preceding 1? — Ludwig V
I think it follows that "0.999...." does not equal 1.There is none. Why do you think there is one or should be one? That's why I think you're misunderstanding. There's no element of the a sequence immediately preceding the limit point. — fishfry
OK. I wondered if it worked a bit more widely than that. I don't think that it would work for sqrt2, since Aristotle could prove that it was "incommensurable" without involving decimals. What about π? I was taught that it was 22/7 or 3.14....?It shows how we get the fraction representation of repeating decimals. — Lionino
For me, this issue has a wider context.
This may be a step too far. But there are many people who turn up on this forum - and elsewhere - who deeply believe that nothing is true and everything is probable. — Ludwig V
The usual basis for this is traditional (since Descartes) scepticism, and one usually tries to meet it by arguing about that.
But what if they have been introduced to probability theory and infinity? Suddenly, there is a mathematical proof.
Sometimes probability = 1 and 1 = 0.9999... So everything is probability, — Ludwig V
I think this is a mistake, because it neglects context. But it is new angle on the mistake. — Ludwig V
I'm basing this on an assumption that both theses are correct - in their context. — Ludwig V
I think it follows that "0.999...." does not equal 1. — Ludwig V
Sadly, my best time for philosophy is first thing in the morning... — Ludwig V
I don't know how you relate that to truth. — fishfry
Just an aside. You probably know this stuff. But others might not. This is not a rigorous presentation. — fdrake
When you talk about the probability of something, that needs to be defined as an event. Which is a particular kind of mathematical object. It does not tend to be the kind of mathematical object that a formula in a mathematical argument is. Eg the probability that it will be raining in 2 hours given that it is raining now makes sense. The probability that 2+2=4 doesn't make too much sense.
However. If a statement A is provable from a statement B and concerns a quantity *, the probability of A given B is 1. As an example, what's the probability of X+1=4 given that X=3? Probability 1.
Another fact like this is that if A and B are mutually contradictory, the probability that A occurs and B occurs is 0. That also works with entailment. Like the probability that X=3 given that X+1=2 is 0, since X+1=2 implies X=1, and there's "no way" ** for X to be 3 given that assumption. — fdrake
The same holds for statements *** you can derive from B using classical logic and algebra and set operations. eg if the probability that X=3 is 0.3, what's the probability that (X=3 or X!=3)? 1, since those are exhaustive possibilities. The latter does have a connection to truth, as if you end up asking for the probability of something which must be true, its probability is 1. — fdrake
For folks like Fishfry, I'm sure you can make the amenable sense I've not specified precise. Logical, algebra and set operations which can be represented as measurable functions on the sample space work like the above. "no way" corresponds to the phrase "excepting sets of measure zero". Which is the same principle that stops you from asking "What's the probability that clouds fly given that x=2?", as there's no way of unifying both of those types of things into a cromulent category of event. — fdrake
The latter also blocks a more expansive connection to truth. Since the kind of things that humans do while reasoning from premises typically aren't representable as measurable functions. Maths objects themselves also have plenty of construction rules that behave nothing like a probability - like the ability to conjure up an object by defining it and derive a theorem about it, there's just nothing underneath all maths that would take take a probability concept which would usefully reflect its structures I believe. — fdrake
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