Per my old topology course, it contains every neighborhood around 1.
One could, if desired, define a new topology that has new rules. But, as leachim noted, it would force us to give up certain properties of numbers and sets, which we’re pretty well dependent on. As he notes, these properties are useful, where a new, proposed, hypothetical mathematics with different rules has not displayed any possible use.
netzweltler is perfectly free to make up his own mathematics, just as Dr. Seuss was free to make up his own alphabet (in On Beyond Zebra, a darn fine exploration of crypto-orthography.) What he cannot do is insist that his definitions are correct, as accepted by mathematicians in the real world.
I’m not a mathematician, but on reading leahcim’s link about the greatest lower bound property and surfing to the Completeness of the real numbers article, I have realized that this line that netzweltler keeps talking about can’t be the real number line at all.
It’s the surreal number line … which includes irrational numbers but not repeating decimals … nor (apparently) the number one … nor any number that can be a limit of any kind … but the most important feature of the surreal number line is that infinity is a very very large finite number … and it is useless … you’re welcome …
So, without this definition (possibly axiomatically defined and therefore “useful”) it is still true to say “If none of the elements in [0, 1) is as close as zero distance to point 1 then the set cannot be as close as zero distance to point 1”, right?
You wouldn’t say “it still has endpoint 1” for any of the finite sets
{ 0, 0.9, 1 }
{ 0, 0.9, 0.99, 1 }
{ 0, 0.9, 0.99, 0.999, 1 }
…
if we remove point 1 from the set, right?
Why do you think it is true for the infinite set [0, 1]?
You’ve confused two totally different things - a set, which is a defined group of things, and a range, which is all the numbers between a high and a low value. What’s more, it doesn’t map to a set unless you do more to define it, such as telling us what set of numbers within that range you want.
And in this case, if the set you’re talking about is {x ∈ ℝ, 0 ≥ x ≥ 1} (all real numbers between 0 and 1), then we run into entirely different problems, because the real numbers are uncountably infinite - meaning you literally cannot get from point A to point B by counting. If you remove 1 from that set, changing it to {x ∈ ℝ, 0 ≥ x > 1}, then you literally cannot define a “greatest number” for the set, making the situation quite different.
Without some definition, it is impossible to say anything about distance between sets and points. That is how math works – you define things and then use those definitions to prove things. There is no, “I don’t want to define this thing, but I am sure I know something about it”.
Again, if you posit a definition for “endpoint” as “the greatest lower bound or the least upper bound of the set” then it is obvious. { 0, 0.9, 0.99, 0.999 } has endpoints 0 and 0.999. [0, 1) has endpoints 0 and 1. Start with the definition, derive the theorem.
If you don’t like that definition, propose another, but you can’t get away without one.
This “work from analogy” thing is great for English common law, but it is not mathematics.
The only comment I wish to add here is to remind netzweltler that the question at hand is the length of a line segment that is defined as having endpoints at 0 and 1 … these endpoint themselves are points and are dimensionless … thus have no part in the evaluation of the length …
Welcome to the Straight Dope Message Boards … if you choose to log in with your pride … you can expect it to be bruised … sometimes it’s best to leave your pride back in Real Life so you can actually learn something new … there’s way too many people on these boards a hell of a lot smarter than you and I combined …
I suppose it’s been pointed out that 0.999~=0.9 + 0.09 + 0.009 + … = 9*Sum[sub]n=1[/sub][sup]infinity/sup[sup]n[/sup], where the sum is of the form Sum[sub]n=1[/sub][sup]infinity/sup[sup]n[/sup], which, for 1/x<1, converges to 1/(x-1), i.e. 1/9 in our case?
Grammering is as overratted as spelting … phaw … I couldn’t be less proud of believing the subject of an subordinating conjunctivitis clauses takes the nominative case … it just sounds better …
Yes. netzwelter’s whole shtick is that no such series sum to an actual number. Not that you can’t complete an infinite sum, mind you, but that such a sum is by necessity undefined and somehow not on the number line …
I believe you’re correct when “than” is considered a conjunction, but the issue is that sometimes “than” is considered a preposition. FWIW, I agree with you in this case.
However, shouldn’t it have been “…there are way too many people…” rather than “…there’s (there is) way too many people…”?
