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Carleas wrote: So the term "the smallest thing remaining countable" is undefined, there can be no smallest real number distinct from zero ("smallest" in absolute value, I assume that is your intent as well).
There is a persistent but unfounded notion that between can be used only of two items and that among must be used for more than two. Between has been used of more than two since Old English; it is especially appropriate to denote a one-to-one relationship, regardless of the number of items. It can be used when the number is unspecified economic cooperation between nations , when more than two are enumerated between you and me and the lamppost partitioned between Austria, Prussia, and Russia — Nathaniel Benchley , and even when only one item is mentioned (but repetition is implied). pausing between every sentence to rap the floor — George Eliot Among is more appropriate where the emphasis is on distribution rather than individual relationships. discontent among the peasants When among is automatically chosen for more than two, English idiom may be strained. a worthy book that nevertheless falls among many stools — John Simon
This is one of those issues that display the clear distinction between a good philosopher and a expert mathematician.
A good engineer will tell them that they are wasting their time.
Reminds me of this joke:
A mathematician and an engineer are sitting at a table drinking when a very beautiful woman walks in and sits down at the bar.
The mathematician sighs. "I'd like to talk to her, but first I have to cover half the distance between where we are and where she is, then half of the distance that remains, then half of that distance, and so on. The series is infinite. There will always be some finite distance between us."
The engineer gets up and starts walking. "Ah, well, I figure I can get close enough for all practical purposes."
Carleas wrote:That phrase is pretty vague
the way you are using it, it is a contradiction of much of standard math, e.g. the associative property of multiplication on the real numbers, the property that the set of real numbers is closed under addition and multiplication, etc
Carleas wrote:Magnus Anderson wrote:\(L + 1 - 1\) is fine.
How can that be? L+1 is undefined! You can't subtract 1 from an undefined quantity and get a defined quantity.
Carleas wrote:Sure, but between \(1\) and \(0\), there are a bunch of other numbers; we can point to any of them (0.123, for example) and prove that \(1\) and \(0\) are distinct numbers. There are no numbers between \(1\) and \(0.\dot9\), which isn't possible if they are distinct real numbers.
Magnus Anderson wrote:What exactly is vague about the statement "A number larger than every other number"?
Magnus Anderson wrote:What makes you think that \(L\) is a real number and that what applies to real numbers also applies to it?
Magnus Anderson wrote:we do have the definition of \(L\). You're merely choosing to ignore this for some reason.
Magnus Anderson wrote:I never confirmed or denied this inequality since it doesn't strike me as particularly necessary.
Magnus Anderson wrote:why do you reject pseudo-numbers such as numbers larger than the largest number?
Magnus Anderson wrote:According to some, \(\infty - \infty = \infty\).
Carleas wrote:\(L\) is clearly not a number like the ones we usually work with, and so I'm not making assumptions about what you mean by "number". You haven't clarified what happens when we subtract 1 from \(L\), what kind of number do we get? If it's a real number, than that number plus 1 must be \(L\), in which case \(L\) is the sum of two real numbers. If it's some number less than \(L\) but of the same type, then does \(L-1 = L\)? Is \(L\) hyperreal?
Here is a great example: just confirm or deny the inequality!
This here is treating \(\infty\) like a number, and it isn't a number. I would say that that equation is meaningless.
When I say that two infinities are equal, I mean that every element of one set can be mapped to exactly one element of the other set. So, for example, \(0.\dot9\) and \(9.\dot9\) have the same number of decimal places, in the sense that each decimal place in one corresponds to exactly one decimal place in the other.
Magnus Anderson wrote:I am not sure why you're insisting on categorizing \(L\). What exactly do we gain by placing it in a category?
Magnus Anderson wrote:I don't have to confirm or deny irrelevant claims.
Magnus Anderson wrote:If I recall correctly, your claim is that \(0.999…=1\). If this is true, it means that you think that an infinite number of non-zero terms (which is what \(0.\dot9\) is) can be equal to a number (such as \(1.\)) In other words, it would imply that at least SOME infinite quantities are numbers (since \(1\) is a number, right?)
Magnus Anderson wrote:\(0.\dot9\) and \(9.\dot9\) have the same number of decimal places because you said so.
Mowk wrote:And the result of dividing .9 recurring in half?
Carleas wrote:Mowk wrote:And the result of dividing .9 recurring in half?
The long division makes sense here, and doesn't really inform the question.
\(\frac{0.\dot9}{2}= .4\dot9 = .5\)
Mowk wrote:Interesting but you are not expressing the fraction .9 recurring. divided by 2 as a fraction.
Mowk wrote:Does it matter how many 9's follow the expression as the expression itself clearly declares there are zero 1s. How can a description of a number that has zero 1's be equal to an expression that has one?
Carleas wrote:Mowk wrote:Interesting but you are not expressing the fraction .9 recurring. divided by 2 as a fraction.
My point was that the division works as expected: the long division algorithm produces \(0.499999...\)
So, I don't think it answers the question either way because if \(0.\dot9 = 1\), then \(0.4\dot9 = 0.5\), and if it doesn't it doesn't.Mowk wrote:Does it matter how many 9's follow the expression as the expression itself clearly declares there are zero 1s. How can a description of a number that has zero 1's be equal to an expression that has one?
There's a lot of addition in there too, and I don't think anyone's denying that 0x1+5/10+5/10 =1
Careleas wrote:Your conclusion doesn't follow. \(0.\dot9\) has infinite decimal places. It is not itself infinite, and infinity isn't used as a number in defining that.
We gain the ability to have a conversation! If your argument depends on the properties of \(L\), then we need a full description of those properties (e.g. by placing \(L\) into an existing category) to evaluate your claims.
Mowk wrote:The question; Is 1 = 0.999...?
Another way of asking the same question might sound something like; is an infinite number of smaller and smaller "parts" equal to 1 finite "whole" ?
If I had 1 volume of space and I divided it, equally in half, an infinite number of times, the result would be an infinite number of parts. I guess if I could cut a volume of space an infinite number of times, it would still have come from the 1 original volume. Seems like the argument could be made from there, that infinity = 1.
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