Finitism and .999....=1 question

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Because it seems to me that the difference between 0.999... and 1 is a matter of how numbers are represented in the decimal system, and not a philosophical or mathematical concept like finitism. In summary, the conversation involves a discussion about whether 0.999... is equal to 1, with one person mentioning the concept of finitism and the other not being familiar with it. The conversation also touches on the ambiguity of decimal representations and the existence of terminating and non-terminating decimal numbers.
  • #1
huginn
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I had a discussion with someone about does .999...=1. She used finitism and I don't know enough about it to reply. I read a little about finitism but I don't know what to make of it, it just seems wrong
Can someone recommend a site with a good rebuttal
 
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  • #3
The shortest version: ##0,999\ldots = 1##.
The short version: The left hand side is a different representation than on the right. So first of all, the left hand side has to be explained. If it is a limit, then it equals ##1##. If it is something else, then what?
The long version: https://www.physicsforums.com/threads/0-999-0.919086/
 
  • #4
##0.99999... = 0.\bar9= 3 \times \frac{1}{3} = \frac{3}{3} = 1##

I don't know anything about finitism. And I also don't know why one needs to be all philosophical about the question.

As I see it the problem just lies in the way numbers are represented in the decimal number system. Nobody would ever argue that ##\frac{3}{3} = 1##. If we used a number system base 60 like on our clocks where ##\frac{1}{3}## would be something like 20 minutes, the question whether 60 minutes is one hour wouldn't come up either.
 
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  • #5
Just to pile on, apparently Finitism sees some academic discussion occasionally, but unless your friend is a research logician, she probably doesn't know as much about it as she thinks she does. The type of finitism that asserts ##0.999... \neq 1## also must assert some weird propositions, like ##N+1\ngtr N## for sufficiently large ##N##.
 
  • #6
I suppose one particular way to see this equality can be to see it as a geometric series (probably already mentioned in thread linked above). For example, set:
a = initial term = 9/10
r = multiplication factor = 1/10

Because r<1 we know that the following sum converges:
a+ar+ar2+ar3+ar4...

S=a/(1-r)=1

Interestingly, a similar equality will seemingly follow no matter number-base system we use.

What is more interesting would be to prove the converse: That is, the only source of ambiguity in decimal representation can occur between:
---- decimal representations with infinite strings of 9's
---- decimal representations with infinite strings of 0's
I think it might be a bit tedious to prove but it could be done by a more careful analysis.

===============

Also I think probably the term ultrafinitism might be more in line with regards to question in OP.
 
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  • #7
SSequence said:
What is more interesting would be to prove the converse: That is, the only source of ambiguity in decimal representation can occur between:
---- decimal representations with infinite strings of 9's
---- decimal representations with infinite strings of 0's
I think it might be a bit tedious to prove but it could be done by a more careful analysis.
Actually, I don't think it would be hard to prove at all. Any decimal fraction (i.e., base-10) that has an expression of the form ##.d_1d_2 \dots n999\dots## can also be represented as ##.d_1d_2 \dots m000\dots##, where m = n + 1. Of course, I'm not proving anything here, but for any given decimal fraction in the first form, you could prove that it is equal to the second form, using the same geometric series argument that you gave.

The ambiguity of these two decimal fractions has to do with 9 being the largest decimal digit. If we were dealing with base-2 fractions, 1 is the largest binary digit, so a binary fraction of the form ##.d_1d_2\dots n111\dots## would be the same as ##d_1d_2\dots m000\dots##, again with m = n + 1, and this could be proved in the same manner. In base-3, we have the ambiguity in trinary fractions of the form ##.d_1d_2 \dots n222\dots##, as 2 is the largest base-3 digit.
 
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  • #8
Yeah, basically I think the most illustrative case would be of:
1=0.999...
I think apart from the equality of these (as is mentioned frequently), it is also important to show that 0.9999... is the only other decimal representation for 1**. I don't think it should be quite difficult (but certainly illustrative). Once one shows that it would be reasonably clear that it can be extended as a more general result.

Furthermore, one would also want to show that apart from certain kind of decimal representations(terminating ones and ones with infinite string of 9's), all other decimal representations uniquely describe a real number.

** Obviously assuming the convention that unnecessary 0's towards the end are cut-off. For example, 1.000 is counted as same decimal representation as 1.
 
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  • #9
Well, how does she define 0,999... in her view?
 
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Related to Finitism and .999....=1 question

What is finitism?

Finitism is a philosophical and mathematical theory that states that only finite objects and quantities can exist. This means that infinite objects, such as infinite sets or numbers, are not considered to be real.

Can 0.999... ever equal 1?

Yes, according to the concept of finitism, 0.999... does equal 1. This is because in finitism, numbers are understood to be finite and therefore, no number can be infinitely close to another number.

How is it possible for 0.999... to equal 1?

It is possible because the decimal representation of 0.999... is an infinite series that approaches the value of 1. In finitism, this infinite series is not considered to be a real number, but rather a finite representation of 1.

Are there any real-world applications of finitism?

Finitism may have implications in fields such as computer science and physics, where the concept of infinity is often used. It can also have implications in the philosophy of mathematics and logic, as it challenges the traditional view of infinitely large and small numbers.

What are some criticisms of finitism?

Some criticisms of finitism include the limitation it places on mathematical concepts and the difficulty in reconciling it with certain mathematical theories, such as calculus. It also challenges the widely accepted notion of infinity in mathematics, which may be difficult for some to accept.

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