Ok never mind…I thought you were referring to the typos in my post not yours
My intuitive answer is “I don’t know what you’re asking, because I don’t know what 1/infinity means.”
No, the difference between .999… and 1 is entirely dependent on the rigorous definition of the relationship between the least member of the reals and the meaning of the ellipsis. Can you rigorously define that relationship?
There is no least member of the reals, so the relationship is one that is vaccuously satisfied by the least member of the reals and any entity, including the ellipsis. (This follows from the fact that an empty description or name satisfies all predicates vacuously.)
Something is wrong here. Did you mean to write .99 the first time and .9 the second? Or did you mean the centimeters where labled and the millimeters were not? Because .999 meters is done to millimeter precision and you appear to have some point relating to the centimeters not being labled. I’ll assume you meant to have a stick labled at centimeter level and not millimeter, since that’s how they’re usually made in the real world.
They are not theoretically co-located. For one thing, if your point is at .999 on the measuring stick and you are for some reason restrained to use centimeter precision despite the measuring stick being more accurate, you would, in the practical world, write down 1.00, not .99, since .999 would be closer to 1 than to .99
And for another, in the practical world, there always exists the concept of a more precise measurement and the concept of measurement uncertainty, so when you write down .99 you’re implicitly indicating your measurement is in the interval .985 and .995.
.999… would have infinite precision and thus be indistinguishable from 1.000…
Yes thanks for the catch…there is an extra 9 in there…it should be .99 was measured and .9 was written down
If it is true that there is no least member is not true that any member can be defined by the fraction 1/10^n where n is unbounded and thus the number of zeros in the decimal representation of 1/10^n is unbounded?
No~ (implying an infinite number of no).
Cantor made infinity rigorous because intuition failed. As soon as you try to use your intuition you wind up with statements like:
That’s a swamp which leads to nonsense answers like 1 = 0. As stated on that page it is undefined (or indeterminate).
You also cannot try to insert physical processes onto infinity. You again get nonsense.
What Cantor did was to throw out intuition. He developed a set theory that allowed him to define every aspect of every term needed and then showed how those sets could be used to make infinities dance. Your give reference to some phrases you call set theory, but, well let’s just say they don’t resemble Cantor’s set theory, to say the least. Cantor’s sets can be explained in words, but it doesn’t work backward. You can’t take words and disprove Cantor. All they show is that you aren’t doing actual math.
This is the heart of the issue with you and all the others who have argued here, not just in this thread but in the many other threads on this issue. You’re not arguing math at all: you’re arguing that your intuition disagrees with the math and you trot out whatever limited arithmetical skills you have to put numbers on your intuition without any definitions, theorems, proofs, or the other apparatus of mathematics. This can’t work. Even if you were a genius, the language of math doesn’t allow for it to work.
FYI, you’re making the same logical fallacy here as “There are no atheists in foxholes”.
Yes.
But I’ll head off what I suspect your reply would be by reminding you that you cannot infer the following B from the following A:
A: the number of zeroes that can appear in decimal expansions of real numbers is unbounded.
B: There is a real number with an unbounded number of zeroes in its decimal expansion.
Huh?
That there is no “least real” is obvious from construction. Propose to me a number which you consider the “least real.” I will promptly divide it by two. Disproof by construction.
And as for “any number can be defined by the fraction 1/10^n,” I simply point to the number 2. 2 = 1/10^n for what n, exactly? There is no counting number n provides a solution to this equation.
(n = the log base 1/10 of 2, but that’s no counting number!)
As far as I can tell, you’re using perfectly cromulent words…but not in any formally valid meaning. It’s like the formulation 0.999…1 It’s a lovely term, but how, exactly, is it defined? “You write down nines forever and then stop and write a one instead.” “Forever…then?” Like “at infinity,” it’s linguistically valid, but mathematically, what does it mean?
I believe his n was meant to range over the reals, not the natural integers.
In that case…what’s the controversy? Log base 1/10 of x = n. What does the “unbounded number of zeroes in the decimal expansion” even mean? Again, he’s using technical language in a way that doesn’t match the conventional definitions.
(In another thread, I mentioned a friend who refuses to believe that the recession is over. Times are still hard and lots of people are unemployed. The trouble is, economists have a specific formal definition for a “recession,” and, in the U.S., that definition hasn’t been met for the past few years. My friend is using his own, personal definition for the word. In the sciences, you can’t do that!)
Heaven only knows why I’m wading in again.
A few points about Valmont314’s lengthy post.
If you are talkng about the integers you need to go back to Peano’s axioms. The distance between any successive numbers is simply defined via an axiom - the successor operation. The fact that the difference between 0 and 1 is the same as the difference between other pairs of successive numbers is not special. It derives from the axioms. It useful because 0 is the additive identity, but that is all. The definition of the rest of the numbers derives from the integers, not the other way around.
You use the term recursive, but never define a base case. Note that there is nothing special about recursion. Any recursive mechanism can be recast as an iterative one with a simple mechanical process. Here this is nothing more than unwinding the recursion and casting the definition of 1/n in its more conventional meaning. If you want to make n go to infinity, just use the successor operator that is defined as one of Peano’s axioms that defined the Integers in the first place. There is nothing special or different about using a recursive definition.
Finally, the core problem is the problem of intuition. Simply, my intuition does not match yours. You can talk about your intuition as long as you wish, but you cannot claim that anyone else’s intuition about the nature of anything is the same as yours. You don’t know what other peoples intuitive sense of the problem is, and most certainly cannot make any claim about the universal applicability or truth of your claims, beyond the scope of your own head. In the end this is why we have formal systems, and formal definitions. Cantor spend a large fraction of his life working out how to get a formal system defined that allowed sensible conversations to proceed about such questions. Without a formal underpinning you may as well be arguing about postmodern interpretations of Jane Eyre.
I’m not sure you understand the confusion I was noting. In your question, you have both a blank randomly in the middle and a “which is:” at the end. Does this mean that, for example, one answer would be:
“The value of the function 1/x when evaluated [b. Finite] is equal to 1/∞ which is: [d. The answer points back to the question]”
?
Jane represents the counting, or “natural” numbers, where Mr. Rochester represents the rationals. Rivers represents the real numbers, especially the irrationals, and, by implication, the unnaturals. Love is seen as a non-transitive operation.
Yes, you are right that
“fill in the blank and answer this question”
Could imply that you were to take one of the multiple choices and use it to fill in the blank…however, I thought it was implicit that the only words that could be filled in were “at infinity” much in the same way that there is only one word that can satisfy the statement “the value of the function y = 2x +3 at _______ is 9” (the only word is three). So I’m not sure where you are going with this but it would seem that you are looking for a degree of rigor in the way the question was formulated that is above and beyond that which is necessary to convey what you were supposed to do.
The point of my post, which clearly was not explained with sufficient rigor as to force binary responses from those responding was to make the following assertions:
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The equation .999… = 1 means that the Real elements .999… and 1 are co-located on the Real number line and for the purposes of measurement are the same point.
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The fact that the elements are co-located on the real number line does not imply that they are the same mathematical object, that is they are still distinct members of the set of Real numbers, otherwise by definition they would not have different symbolic representations.
This distinction can be observed by the fact there is a value for 2 x 1 whereas there is no known value for 2 x .999… -
This would not be the case (ie there would be exactly 0 co-located elements) if the precision with which real numbers were allowed to be represented in decimal form was limited such that .0(sub n)1 = .X (sub n +1) where X represents the number of decimal places in the decimal representation and 0(sub n) represents the number of zeros and n is any positive integer
If that were true either 2 or 0.9999… is not a member of the reals.
Since both are members, there must be such a value. How about 1.99999…
You can say that either 2 is the integer 2, in which case we can define 2 x 0.999… as:
0.999… + 0.999… , and so on for x * 0.999… for n a member of the integers. Then sum the terms starting from the left.
or you can take the general case, of x * 0.999… where x is a member of the reals. Then you just apply the multiply with carry rules starting from the left. (See an earlier post about starting from the left.)
Yes, “there is a value for 2 x 1” – it is 2. In other words, “2 x 1” and “2” are “different symbolic representations” of the same real number.
Therefore, the fact that “1” and “.999…” are “different symbolic representations” does not imply that they are distinct real numbers.
Please try to make your logic consistent!