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mfb wrote:>> Thus the value of the index must become infinite
No. There is an infinite number of finite integers. You just can't give an upper bound for your integers (because there is none).
Given the way I've ordered the set, the number of elements in the set cannot exceed the values that are allowed to exist within the set, because each element of the set is a value corresponding to exactly how many elements are included up to that point.
arbiteroftruth wrote:mfb wrote:>> Thus the value of the index must become infinite
No. There is an infinite number of finite integers. You just can't give an upper bound for your integers (because there is none).
Perhaps I should phrase it this way:
Given the way I've ordered the set, the number of elements in the set cannot exceed the values that are allowed to exist within the set, because each element of the set is a value corresponding to exactly how many elements are included up to that point. So if the number of elements within the set is infinite,
then the set must contain elements with a value of infinity.
arbiteroftruth wrote:Perhaps I should phrase it this way:
Given the way I've ordered the set, the number of elements in the set cannot exceed the values that are allowed to exist within the set, because each element of the set is a value corresponding to exactly how many elements are included up to that point. So if the number of elements within the set is infinite, then the set must contain elements with a value of infinity.
arbiteroftruth wrote:But then, and this is switching topics, that makes me wonder something else about various infinities. The only proof I know of(and there may well be more for all I know) that the real numbers are uncountable while the integers aren't goes something like this. Try to make a list of all the real numbers, and this will be a list of infinitely long strings of digits. I can then take the first digit of the first element in the list, and specifically choose a different digit for that position. I can then take the second digit of the second element, and again specifically choose a different digit for that position. Thus, I can guarantee that there is no element in your list that matches all the digits I have selected, because I have specifically chosen digits to disagree with every element on the list in at least one position. Thus, my digits constitute a valid number that isn't on your list, therefore the initial assumption that you have a list of all the reals has created a contradiction, thus such a list is impossible.
But can't the exact same argument be made for, for example, the integers? I can select a ones digit that disagrees with the first element in your list, a tens digit that disagrees with the second element in the list, and so on, thus guaranteeing that I generate a number that isn't on your list.
arbiteroftruth wrote:And to come at it from the other direction, as I understand it we prove that the set of rational numbers is no greater than the set of integers by finding a method that assigns each rational number to a corresponding integer and guarantees that no rational numbers are missed. Thus we conclude the set of rational numbers cannot be greater than the set of integers.
But can't the same be done for the reals? If every real number can be expressed as an infinite decimal, then I can equate the integers' ones digit to the reals' ones digit, the integers' tens digit to the reals' tenths digit, the integers' hundreds to the reals' tens, the integers' thousands to the reals' hundredths, and so on, and guarantee that all possible digit positions receive all possible values, thus no real number is left out.
arbiteroftruth wrote:Try to make a list of all the real numbers, and this will be a list of infinitely long strings of digits.
arbiteroftruth wrote:But can't the exact same argument be made for, for example, the integers? I can select a ones digit that disagrees with the first element in your list, a tens digit that disagrees with the second element in the list, and so on, thus guaranteeing that I generate a number that isn't on your list
ahammel wrote:You can't construct an integer from the diagonal of an infinite matrix because integers have a finite number of digits by definition.
Same problem: you can't have infinitely long integers, so you can't map the integers to the reals like that.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
pollywog wrote:I want to learn this smile, perfect it, and then go around smiling at lesbians and freaking them out.Wikihow wrote:* Smile a lot! Give a gay girl a knowing "Hey, I'm a lesbian too!" smile.
arbiteroftruth wrote:ahammel wrote:You can't construct an integer from the diagonal of an infinite matrix because integers have a finite number of digits by definition.
Same problem: you can't have infinitely long integers, so you can't map the integers to the reals like that.
That restriction on integers seems arbitrary though. What contradiction, if any, would arise if I defined a class of objects identical to the integers except that the number of digits is allowed to be infinite? And if there is no contradiction, why do we build that restriction into the definition?
EDIT: Come to think of it, there's no way a contradiction could come out of that, as the set I'm thinking of already exists. It's simply the set of all real numbers in which every digit right of the decimal is 0. So why don't we allow the set of integers to be equal to that set?
jestingrabbit wrote:What would fail is mathematical induction ie there would be propositions that are true for 0 and true for n+1 if true for n, but not true for all integers (a trivial example of such a proposition would be "having a finite number of digits in its representation", but other problems, like determining divisibility or primality, would also crop up pretty quickly).
arbiteroftruth wrote:jestingrabbit wrote:What would fail is mathematical induction ie there would be propositions that are true for 0 and true for n+1 if true for n, but not true for all integers (a trivial example of such a proposition would be "having a finite number of digits in its representation", but other problems, like determining divisibility or primality, would also crop up pretty quickly).
Would not my original post be an example showing that that's already the case? If the proposition is "the set of integers from 0 to n has exactly n+1 elements in it", this is true when n=0, and true for n+1 whenever it's true for n, but is not true of the set of all integers.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
arbiteroftruth wrote:Come to think of it, there's no way a contradiction could come out of that, as the set I'm thinking of already exists. It's simply the set of all real numbers in which every digit right of the decimal is 0. So why don't we allow the set of integers to be equal to that set?
ahammel wrote:You can't apply the diagonal argument to that set because as soon as you change one of the zeros trailing the decimal to something else, the result is a number that isn't in the set.
arbiteroftruth wrote:Similarly, I'm not sure what you mean either. Could you clarify your objection?
You wrote: "If the proposition is 'the set of integers from 0 to n has exactly n+1 elements in it'...." This is a proposition about the natural number n.arbiteroftruth wrote:Similarly, I'm not sure what you mean either. Could you clarify your objection?
arbiteroftruth wrote:ahammel wrote:You can't apply the diagonal argument to that set because as soon as you change one of the zeros trailing the decimal to something else, the result is a number that isn't in the set.
True, but it doesn't have to be a perfect diagonal. To use the standard example of the reals, the example can be easily modified to not include every digit. I could make my first digit disagree with the first digit of the first element, pick my second digit completely at random, make my third digit disagree with the third digit of the second element, pick my fourth digit at random, and so on. Or I could simply pick the first n digits at random, make my n+1th digit disagree with the n+1th digit of the first element, and so on.
Similarly, if using the set of all reals in which all digits right of the decimal are 0, I simply leave all those positions alone, and start taking a "diagonal" from the ones digit leftward. The result is guaranteed to not match any existing element in the list, and still satisfies the definition of the set in question.
arbiteroftruth wrote:I suspect I'm missing some simple clause about the way we define things like "infinity", but this apparent paradox occurred to me.
Suppose I want to make a set of all the finite positive integers. I order them as 1, 2, 3, ..., such that each element is necessarily equal to its index within the set.
Since there is no upper bound to finite integers, the size of the set must be infinite. Thus the value of the index must become infinite.
ahammel wrote:No it isn't. If you leave element n of the diagonal unchanged, the number constructed from the diagonal isn't guaranteed to be different from the real number at n.
arbiteroftruth wrote:Ah, I see my problem here. My usual approach to the behavior of infinity in various circumstances is to take a calculus approach, examining the behavior of the scenario as the variable of interest becomes arbitrarily large, and defining the behavior at infinity to be the same as the limit found by increasing the variable.
fishfry wrote:arbiteroftruth wrote:Ah, I see my problem here. My usual approach to the behavior of infinity in various circumstances is to take a calculus approach, examining the behavior of the scenario as the variable of interest becomes arbitrarily large, and defining the behavior at infinity to be the same as the limit found by increasing the variable.
Then your approach to calculus will run into trouble too. Example: Take a sequence whose n-th term is 1/n. So the sequence is 1, 1/2, 1/3, 1/4, ...
Now it's certainly true that each element of the sequence is strictly greater than 0.
Is that true for the limit of the sequence? No, it's not. Properties frequently are NOT preserved by taking limits. So if you're carelessly just assuming that limits preserve all properties you're interested in, you'd get into a lot of trouble.
A more striking example is to start in the lower-left corner of a square and go one unit to the right, one unit up. Total path is 2.
Now instead, start at the lower left and go 1/2 unit right, 1/2 unit up, 1/2 unit right, 1/2 unit up. Now your total path is still 2. If you keep on doing this you'll see that your path appears to converge to the diagonal, which has length sqrt(2). But the limit of the "little to the right, little up..." path is 2.
You can NOT blindly take limits and assume that properties are preserved.
arbiteroftruth wrote:Similarly, I'm not sure what you mean either. Could you clarify your objection?
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
++$_ wrote:You wrote: "If the proposition is 'the set of integers from 0 to n has exactly n+1 elements in it'...." This is a proposition about the natural number n.arbiteroftruth wrote:Similarly, I'm not sure what you mean either. Could you clarify your objection?
Now, the set N of all natural numbers is not of the form "the set of integers from 0 to n." (Other sets that are not of this form include the set of rational numbers, the set of real numbers, and the set of one-legged barmaids in Switzerland.)
So why would you think your proposition was true about the set of all natural numbers? It doesn't even make sense for that set, because the quantity "n+1" isn't defined when you haven't got an n.
To put it another way, you certainly wouldn't claim it's true of the set of all one-legged barmaids in Switzerland, would you? And if you did, what would it even mean?
arbiteroftruth wrote:I haven't rigorously proved this of course, but it seems to me...
Talith wrote:arbiteroftruth wrote:I haven't rigorously proved this of course, but it seems to me...
This is where you should have stopped. Think about these things of course, play about with them, enjoy the head aches, but be careful with statements like the above; they always tend to bite you in the ass.
arbiteroftruth wrote:Talith wrote:arbiteroftruth wrote:I haven't rigorously proved this of course, but it seems to me...
This is where you should have stopped. Think about these things of course, play about with them, enjoy the head aches, but be careful with statements like the above; they always tend to bite you in the ass.
I don't see how it possibly could, since an admission of uncertainty is built right into the structure of the sentence. But fine, I acknowledge that I have been warned.
pollywog wrote:I want to learn this smile, perfect it, and then go around smiling at lesbians and freaking them out.Wikihow wrote:* Smile a lot! Give a gay girl a knowing "Hey, I'm a lesbian too!" smile.
ConMan wrote:Also, you're proposing a system where you create awkward redefinitions of objects in order to avoid awkward results - which tends to go against lots mathematics where the aim is to start with simple constructs and see what, ugly or elegant, can be proved from them. As a result, you're going to have to ask a very difficult question - what *can't* your system do that existing systems can't, because by removing some of the counterintuitive results you're also almost certainly going to remove some more useful (or even important) ones.
The set of integers has no upper bound, but each element in that set is finite. Is this distinction still not clear to you?arbiteroftruth wrote: at least find that quirk of my system no more objectionable or counterintuitive than the standard system's quirk that integers simultaneously have no upper bound yet are by definition finite.
gmalivuk wrote:The set of integers has no upper bound, but each element in that set is finite. Is this distinction still not clear to you?arbiteroftruth wrote: at least find that quirk of my system no more objectionable or counterintuitive than the standard system's quirk that integers simultaneously have no upper bound yet are by definition finite.
arbiteroftruth wrote:gmalivuk wrote:The set of integers has no upper bound, but each element in that set is finite. Is this distinction still not clear to you?arbiteroftruth wrote: at least find that quirk of my system no more objectionable or counterintuitive than the standard system's quirk that integers simultaneously have no upper bound yet are by definition finite.
It's perfectly clear and has been for some time. It's just at least as odd as the result I described within my hypothetical system.
Fine, but that only gets you out of that one particular paradox. What about the property, "is even (x)or odd"? 1 is even xor odd, 2 is even xor odd, ... n is even xor odd, but their limit is neither. In fact, I can think of very few typical properties of integers that could possibly be applicable to infinity. "Is 0 or 1 or strictly less than its square", "is 0 or has a unique prime factorization (modulo sign)", "is congruent to K modulo N" for any integer choice of K and any N with |N|>1, and on and on and on.arbiteroftruth wrote:That's a matter of properly defining the operation of taking the limit. That is, by definition, taking the limit as the variable approaches infinity implies explicitly replacing the property "is finite" with the property "is infinite"
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