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Magnus Anderson wrote:4) \(999\dotso\) has no explicit digit associated with \(10^0\) but it has an implicit one and that digit is \(0\).
The representation 999... refers only to the digit "9" being repeated endlessly.
There is no \(0\) digit involved in 999... implicit or otherwise.
Magnus Anderson wrote:Alright, you stated that you disagree with (4).
Are you going to present an argument against (4)?
Or perhaps you think that this . . .The representation 999... refers only to the digit "9" being repeated endlessly.
There is no \(0\) digit involved in 999... implicit or otherwise.
. . . is an argument against (4)?
Magnus Anderson wrote:Are you saying that proving a negative is an impossible task?
I thought you don't subscribe to such popular but nonetheless erroneous ideas.
obsrvr524 wrote:I didn't say that it is impossible. I said that when someone simply says that something is there and we do not see it there  there is nothing to say but  "It is not there  so show us why you think it is there".
If you insist that it is there without being able to provide proving evidence  we jump down on the chart to a categorical disagreement assignment ("Responder Expires" on the next to last flowchart line)  meaning a new class of people who can choose whichever branch of the debate they prefer to believe. Future debates can take it up from there.
Magnus Anderson wrote:obsrvr524 wrote:I didn't say that it is impossible. I said that when someone simply says that something is there and we do not see it there  there is nothing to say but  "It is not there  so show us why you think it is there".
How about presenting an argument that concludes with "Therefore, it is not there"?
Magnus Anderson wrote:One way to argue against (4) is by arguing in favor of your belief that the digit associated with \(10^0\) in \(999\dotso\) is \(9\).
The Initial Argument
(Version 4)
1) Let \(X\) be a class of decimal / base10 numerals that have no fractional part and that stand for a positive number.
2) Any two decimal expressions that are instances of \(X\) represent one and the same number if and only if every explicit and implicit digit in one expression is the same as the explicit or implicit digit located at the same place in the other expression.
3) \(999\dotso\) is an instance of \(X\).
4) \(999\dotso\) has no explicit digit associated with \(10^0\) but it has an implicit one and that digit can be \(0\).
5) The decimal expression of \(\sum_{i=0}^{i>\infty} 9\times10^i\) is an instance of \(X\).
6) The decimal expression of \(\sum_{i=0}^{i>\infty} 9\times10^i\) has an explicit digit associated with \(10^0\) and that digit is \(9\).
7) Therefore, \(999\dotso\) and \(\sum_{i=0}^{i>\infty} 9\times10^i\) represent two different numbers.
Certainly real wrote:Longdivision algorithm is a term I am hearing for the first time. I'm also pretty sure we were never taught exactly why a fraction like 1/3 amounts to .333... . I'm from the UK. Maybe this is something they don't teach us here, or maybe it's just something they didn't teach me at my high school. Alternatively, they taught me and I forgot (sorry if this is true).
Certainly real wrote:Given your use of finite in the above quote, the only way I can see this hold, is if we said that pi is also a finite number.
0 1 2 3 4 5 ...
^

pi
Certainly real wrote:But according to my quick session of google, pi is not a finite number.
Certainly real wrote: But I think I get where you're coming from and I still think it semantically inconsistent to say the nature of .333... is finite and yet equal to 1/3 at the same time.
Certainly real wrote:I understand the move behind saying when you have an infinity of 3s after 0. , you have 1/3.
Certainly real wrote:I am questioning whether this is a semantically justified move.
Certainly real wrote:Indeed, if we could have an infinity of 3s,
Certainly real wrote: it would follow from this that we have absolutely hit 1/3. It’s complete, it’s finished, so 1/3 has been hit.
Certainly real wrote:Yes I do. However, changing to base 3 to say 1/3 = 0.1, to me, is just a different way of representing the semantic of 1/3 of 0.3. It doesn't really change anything semantically as far as I can see, and my issue here is one in terms of semantics, not representation.
Magnus Anderson wrote:1) Let \(X\) be a class of decimal / base10 numerals that have no fractional part and that stand for a positive number.
2) ... (not relevant here)
3) \(999\dotso\) is an instance of \(X\).
Magnus Anderson wrote:I've realized something. (4) is wrong though not entirely wrong. It is possible for the digit associated with \(10^0\) in \(999\dotso\) to be \(9\), it is merely not necessarily so. The key insight is that the position of the first digit in \(999\dotso\) is not specified. It can be literally any. (I think it was phyllo who hinted at something similar in the past.)
Magnus Anderson wrote:For example, if the index of the first digit is \(0\), which means that its weight is \(10^0\), then the digit associated with \(10^0\) is \(9\). The only problem is that it is not necessarily so. For example, the index of the first digit can also be \(infA^2\), in which case, if the number of \(9\)s is \(infA\), the digit associated with \(10^0\) would be \(0\) since the endless chain of \(9\)s would never reach that position.
Magnus Anderson wrote:(There's another problem too. If the weight of the first digit is \(10^0\) then what is the weight of the next \(9\)? There must be "the next \(9\)" since we're dealing with an endless series of \(9\)s in the direction of the least significant digit. It should be \(10^{1}\) but that would mean that \(999\dotso\) has a fractional part which we have previously agreed that it does not have. Thus, in such a case, the expression would represent a conceptually impossible number. And even if it were not a conceptually impossible number, it still wouldn't be \(\dotso999\) simply because
Magnus Anderson wrote:With all that said, I have to revise my argument. Like so:The Initial Argument
(Version 4)
1) Let \(X\) be a class of decimal / base10 numerals that have no fractional part and that stand for a positive number.
2) Any two decimal expressions that are instances of \(X\) represent one and the same number if and only if every explicit and implicit digit in one expression is the same as the explicit or implicit digit located at the same place in the other expression.
3) \(999\dotso\) is an instance of \(X\).
4) \(999\dotso\) has no explicit digit associated with \(10^0\) but it has an implicit one and that digit can be \(0\).
Isn't this a fundamental problem?Your index i cannot exceed infA because it has to remain countable  the natural numbers. The "..." indicates a countable natural number series only.
phyllo wrote:Isn't this a fundamental problem?Your index i cannot exceed infA because it has to remain countable  the natural numbers. The "..." indicates a countable natural number series only.
If infA is infinite, then it's not countable.
If infA is countable, then it's a natural number  not infinite.
If it's an infinity, then you also can't count to it.Since infA is an infinity, the index certainly can't count beyond it.
phyllo wrote:If it's an infinity, then you also can't count to it.Since infA is an infinity, the index certainly can't count beyond it.
And you also can't count almost to it ... ie infA1 would seem to be logically nonsensical.
That's why standard math uses the concept of limits.
obsrvr524 wrote:So I have to still disagree with (4).
I still cannot identify an implicit 0 digit possible anywhere
obsrvr524 wrote:
 If there was an implicit 0 digit there, I would see it
 I don't see an implicit 0 digit indicated.
 Therefore there is no implicit 0 digit present.
Magnus Anderson wrote:obsrvr524 wrote:So I have to still disagree with (4).
I still cannot identify an implicit 0 digit possible anywhere
There are implicit \(0\)s everywhere. For example, there's an implicit digit right before the first digit in \(99\dot9\). That digit is \(0\). There is also an implicit digit at \(1\) (which is the index of the first digit after the decimal point.) That digit is also \(0\).
Magnus Anderson wrote:I assume what you're saying here is that you do not think that the digit associated with \(10^0\) is \(0\) but rather \(9\).
Magnus Anderson wrote:Can I assume that the following argument of yours is applicable to Version 4 of my argument?
https://www.ilovephilosophy.com/viewtop ... 0#p2804739obsrvr524 wrote:
 If there was an implicit 0 digit there, I would see it
 I don't see an implicit 0 digit indicated.
 Therefore there is no implicit 0 digit present.
You said that you still disagree with (4), so it should be applicable.
Magnus Anderson wrote:If it's applicable, I am ready to present my agreements/disagreements as well as counterargument.
Can you explain this distinction?So "infA−1" is technically countable  just not identifiable.
phyllo wrote:Can you explain this distinction?So "infA−1" is technically countable  just not identifiable.
phyllo wrote:That doesn't explain why you think it's "technically countable".
If you can count it, then you can identify it. Countable and identifiable seem to go together.
Wikipedia wrote:Countable
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set.
obsrvr524 wrote:
 If there was an implicit 0 digit there, I would see it
(Disagree.) I don't see an implicit 0 digit indicated.
(Agree.) Therefore there is no implicit 0 digit present.
(Disagree, of course.)
I don't think that you are using it in that sense. Basically by writing infA1, you're counting backwards from infA. That seems like a no no. You know, counting back from where exactly??Wikipedia wrote:
Countable
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set.
Magnus Anderson wrote:The text in red is inserted by me.obsrvr524 wrote:
 If there was an implicit 0 digit there, I would see it
(Disagree.) I don't see an implicit 0 digit indicated.
(Agree.) Therefore there is no implicit 0 digit present.
(Disagree, of course.)
I agree that the conclusion follows from the premises and I agree with the second premise. What I disagree with is the first premise. That said, it is up to me now to present an argument against it. Wait for it
1) What is there is not necessarily seen.
2) Therefore, if there was a \(0\) associated with \(10^0\) in \(99\dot9\), obsrvr524 would not necessarily see it.
Take THAT!
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