That’s really careless. Dismissing the definition of “quantity”? Really? Well then, I guess we can put in whatever we want for n. How 'bout “cow”? Is “cow” > 0? If so, then I can prove that (\sum_{i=1}^{cow} \frac{9}{10^i} < 1). But I think even you can appreciate that there is a limit to what we can substitute for n.
I’m going to demonstrate how you’re confusing intuition for logic. Your argument is:
(\sum_{i=1}^{n} \frac{9}{10^i} < 1) for every (n > 0)
^ Seems logical, right? But you’re missing a crucial step between 2) and 3): (\infty) is a valid value for n. I know you feel intuitively that (\infty) must be a valid value for n because (\infty) is a quantity, so no need to prove it. But when you’re arguing with someone who disagrees with you on that, you do need to prove it. You can’t just run on intuition. If I argued that “cow” > 0, and therefore the formula applies to cows as well, you would insist I prove that “cow” is a valid value for n, wouldn’t you? Same onus falls on you to prove that (\infty) is a valid value for n. But so far, all I’ve seen from you is re-assertion after re-assertion that (\infty) is a number–no proof–which tells me you believe it on intuition, not logic. You need to prove this just as much as you need to prove that what applies to finite sets also applies to infinite sets (and I wonder if this is just a special case of the same thing). But seeing as how you refuse to prove your point when I ask you too, I’m guessing you’ll cower away from this one too.
Well, it applies to any real number greater than 0. But what that means is: grab any number on the number line. (\infty) is not on the number line. It’s a direction in which the number line extends.
That’s a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.
I can have a variable “U”(unicorns) but it does not mean that unicorns are a part of mathematics.
Humans exist within time and therefore need time to understand things.
The explanation I gave earlier in the thread for any “concept of time in mathematics” was that there’s a difference between quantity and representation of quantity. The former is “already there” (for infinite quantities, it’s undefined where exactly “already there” is), but representations of quantities as well as representations of how to construct/deconstruct quantities have ordered steps that occur before or after other ordered steps - according to that order and therefore implying temporality.
When I say endlessness in reference to quantities, I’m referring to the quantitative boundaries (ends) that specfically define a quantity as distinct from higher or lower quantities. Not having this, as with an undefined/indefinite/infinite, means it has no ends. Mathematical construction of such endlessness would take an endless amount of time to complete, just as its deconstruction to a definite quantity would never happen. The best you can do is imply endlessness by using a symbol/notation that only looks like it’s “bounding” boundlessness, which has to be treated very carefully and separately from defined, bounded finites, which have specific ends. Otherwise you can be fooled into thinking there’s more than one kind infinity, and/or that each one can have a different size, when in fact it’s the finite constraints around infinity only that affect anything to do with size: there’s only one way in which endlessness can be endless.
If you are presented with the fraction 1/3 , then it takes some time for you to do the division. It might take you 10 minutes, it might take you 5 seconds to realize that 1/3=0.333… (Or you might never realize it.)
It takes you time because time applies to you. Mathematics is not a “being” who has to “do” something. It doesn’t “need” to “get to the end”. It doesn’t need a process to “do” the calculation. Mathematically 1/3=0.333… - that’s it.
Phyllo, you don’t understand what I’m saying when I say the implication is that all finite numbers are infinite (in fact you avoided it)
Let’s say we hypothetically live in a world where fractions don’t exist… it would be unfathomable that 0.999… = 1.
In a decimal world. 0.111… * 9 equaling 1 is impossible.
The problem is not with my logic, the problem is how operators work with 1 minus base, to make them ‘appear’ equal… but then again, they don’t appear equal at all do they !!!
There seems to be a lot of confusion here about what “infinite” means. “All finite numbers are infinite” doesn’t make any sense as a statement.
That’s simple enough. All you need to do is to restrict yourself to whole numbers, natural numbers or integers.
There is no integer which represents 1/3 or 1/2 or 8/9. When evaluated, those fractions are equal to :
1/3=0 in integer
1/2=0 or 1/2=1 in integer
8/9=0 or 8/9=1 in integer
The two values given for 1/2 and 8/9 depend on whether truncation or rounding(up/down) is the standard procedure when evaluating the results.
A similar thing happens with 0.999… when using real numbers. It “jumps” up to 1.
Yes but in as far as it pertains to quantity, its pertaining makes quantity indefinite.
We agree on that.
Thanks gib.
Yes and I have noticed during the years that the main problem when discussion infinity is its indefinite-ness.
This is the hot core all these debates centre around and it not being made explicit perpetuates the confusion.
How are you not going to be confused if you want definite results out of something that is by definition indefinite?
Doesn’t matter how you word it… you’re like a squirrel running from slingshots right now. You’ll dodge for a while, but, more likely than not, one will eventually connect! To make this rated g, the pebble never hurts the squirrel
I am not saying “This is what infinity is”. I am saying “If this is what infinity means, then this is what follows”. My point being that if (\infty) refers to a specific number that is greater than every integer, then (\infty - \infty = 0) is true but (\infty + 1 = \infty) is not; and if it refers to a non-specific number that is greater than every integer then (\infty + 1 = \infty) is true but (\infty - \infty = 0) is not.
Yes, but in such a case, (\infty + 1 = \infty) is not true.
If (\infty) refers to a non-specific number greater than every integer (in the same way that “A number greater than (3)” refers to a non-specific number since it can be (4), (5), (100) or (1000)), then (\infty + 1) equals to (\infty) (because “A number greater than every integer + 1 = a number greater than every integer”) but (\infty - \infty) is indeterminate.