Do you think the human notion of 'infinite'/'infinity' is actually real infinity?

[HorseshoeHermit]

There are infinitely many primes.
.^. Infinity is real.

What are you on about?

Onejay, could you point out one of the conflicts you name in post 54?
Could you define what 0.9[bar diacritic] is? You are handwaving the existence of an "infinitesimal remainder".

 

[Borachio]

Good grief. Where did you get that? I am stating that calculus is the art of "close enough".J

I got it from your claim that 0.9999... isn't equal to 1; that no matter how many 9s you add to the end of 0.9 you'll never reach 1. That, it seems to me, is exactly what Zeno maintains.

I can understand why you rebel against the mathematical logic of it, though. Maths very often does turn up counter-intuitive results.

 

[Kyriakos]

Good grief. Where did you get that? I am stating that calculus is the art of "close enough".



You don't. Like "1" and "+" it is undefined. If you prefer, you could call it axiomatic. Some things are irreducible.

BTW We have had all this discussion, but no one has ever challenged, or even addressed my proof. Instead you offer flawed proofs that depend on an epsilon. That is a you-first argument. I went first. Your turn.

J

I think that 'irreducible' is not correct (at least in regards to '1' or '+'). While both are used axiomatically, there is little to make one think they have to be in the future as well. For starters '1' as a concept is the topic of thousands of pages even back in Plato's time

However they may be indistinct even when broken up to many levels. Likely remain indistinct regardless of how many levels are added to define them, and math accordingly extrapolated with the effect of those levels as well. In the end something will always be axiomatic in any set system, cause that is the ground sustaining the observer looking around in that system
Not as much a solid ground, as an elevator platform, in some vast mining complex.

 

[warpus]

I got it from your claim that 0.9999... isn't equal to 1; that no matter how many 9s you add to the end of 0.9 you'll never reach 1. That, it seems to me, is exactly what Zeno maintains.

You can't add any more 9s to 0.999...

It's an infinite amount of 9s already

 

[Borachio]

But I expressly stated: "how many 9s you add to the end of 0.9".

If you add an infinite number of 9s (an impossible task, since among other things there's no such thing as an infinite number, but never mind) to 0.9 you end up with 1. That's the point.

Zeno says that no arrow can ever reach its target because of the same reasoning that tells us that 0.9999... doesn't equal 1. But, surprisingly perhaps, arrows aren't ancient Greek philosophers and they get there nonetheless.

 

[warpus]

No amount of 9s you add to 0.9 will give you 1, but 0.999... is equal to 1.

It might seem like a paradox, but it's just a bit counterintuitive, that's all.. Like you said earlier, it does confuse plenty of students. You'll never reach 0.999... by adding 9s to 0.9, so you just can't think of it like that.

 

[Borachio]

An infinite amount of 9s does the job nicely. Arrows do reach their targets.

 

[Kyriakos]

Re Zeno, an approach of his thinking can be:

-Let's suppose that our senses show us something at least close to the truth, and so that there 'really' exist many things (and not just One)
-if Many things exist, then they must be distinct and not a continuum
-If they are distinct then they must have edges
-If they have edges which are non-zero parts, then those edges also have edges
-if their edges include zero-parts then things existing also include non-existence
-the human senses obviously make us sense things as Multitudes, but if those multitudes are rounded up from endless division and including zero-parts, then our senses are clearly not translating the world correctly

(but the above is just a reflection on his statements salvaged as the paradoxa presention of 6,7 and 8, and it is not that thorough... In my thread on Zeno i went on about the work a lot more, link earlier in this thread )

 

[warpus]

An infinite amount of 9s does the job nicely. Arrows do reach their targets.

You won't be able to add an infinite amount of 9s to 0.9

You can start, but you'll never ever finish. It's impossible. That's the point I was making.

 

[onejayhawk]

An infinite amount of 9s does the job nicely. Arrows do reach their targets.

An infinite number of 9s still leaves you on the open segment. 1 is defined to be outside the segment.

The hypothesis fails.

J

 

[Borachio]

You won't be able to add an infinite amount of 9s to 0.9

You can start, but you'll never ever finish. It's impossible. That's the point I was making.

So. You're saying that arrows never reach their targets?

 

[Kyriakos]

So. You're saying that arrows never reach their targets?

If movement doesn't exist, they do not even have a target

You can compare Zeno and Aristotle on their view on whether there is any 'position' where an object is, and therefore can leave that position to go elsewhere. Zeno argues that the position has to be either dependent on the object, or not, and if it is dependent on the object it is not set (supposing the object moves). If it is independent of the object then it is not defined with the help of the object and so would belong to some group of positions. But those may need further over-positions to grant them the quality of 'position'.
Aristotle argues that a position an object is at can be likened to "a vessel the empty space of which the object occupies", and so the position is examined as an object, being in some other position, all the the way to a "prime force of movement, itself immobile". Although i haven't read much on that (mostly it is elaborated on the massive final book of his Physics lectures), it seems certain that he ultimately just attributes stuff to some ideal (but singular) thing of other type, that itself is without such elements, and unknown.

Given that Socrates somewhat tried to give a similar answer to Parmenides (speaking of some ideal and indistinct prime cause of other kind), it is true that Aristotle is not really taking part in the dialectic side of things, and his views would be dismissed by the Eleatics and Plato. But he has a different end anyway, which is to have scientific orders (eg physics and biology) be distinct from theoretical philosophy and philosophy about what notions themselves mean...

 

[BenitoChavez]

Your proof does not work. The first line does not subtract evenly from the second line as shown in the third line. There is an infinitessimal remainer, which you simply dropped.

As you say, the proof is trivial. 0.999... is an element of the set of points in the open unit segment. 1 is not an element, by definition.

Lets take the 1st 2 lines of his proof.

x = 0.999... | * 10
10x = 9.999... | -x

First subtract the left hand side. 10x - x = 9x. I hope we can agree on that.
Now for the right hand side. 9.999... - 0.999... = 9 + 0.999... - 0.999... = 9 + 0 = 9
The result is the third line of his proof
9x = 9

There is no infinitesimal remainder unless you somehow contend that 9 + 0.999... != 9.999... or that 0.999... - 0.999... != 0. I'm not sure what you're on about with unit segments.

If you accept that this is correct. However, it is not.

Assume 1/9 = 0.1111...

9x(1/9) = 9x0.1111... = 1 #

We have proof by contradiction that .1111... =/= 1/9.

J

.111... does equal 1/9. A simple (but tedious) way to see this is to do the long division of 9 into 1 by hand. It should become readily apparent that the pattern isn't going to break.

 

[Kyriakos]

^Intuitively it seems suspect to add an integer to something ongoing and with no end.. I mean imagine if you tried to fuse highly volatile chemical elements with something virtually stable.
And in math the integer (at least currently) is not just virtually defined/stable. It is entirely so.
Furthermore making the integer non-stable would alter the already non-stable other types of numbers or progressions, based on a stable integer set.
(which, btw, i think will have to happen in the future).

 

[nc-1701]

Is .99999.... = 1?

Of course not, but close enough for practical purposes. The proof is trivial. .9999... is on the unit segment open at 1, but 1 is not. QED

That said you will see proofs that they identical , silly as that is. You will even get the statement that an open segment has no endpoint. The correct statement is that an open segment has no rational endpoint. We have the concept of an infinitely small distinction, but handling it is difficult. Instead we approximate, which destroys the purpose.

J

Except that you haven't since the bolded portion of your proof is false.

Consider the open set (0,1), since we are in a metric space this means that for every point x in (0,1) there exists an open ball centered at x wholly contained by (0,1).
Now suppose x=0.99... and x is in (0,1), then for some epsilon the open interval (x-epsilon,x+epsilon) is wholly contained in (0,1), however 0.99...+epsilon > 1.
Therefore either (0,1) isn't an open set (it is) or 0.999... isn't contained in it.

Good grief. Where did you get that? I am stating that calculus is the art of "close enough".



You don't. Like "1" and "+" it is undefined. If you prefer, you could call it axiomatic. Some things are irreducible.

BTW We have had all this discussion, but no one has ever challenged, or even addressed my proof. Instead you offer flawed proofs that depend on an epsilon. That is a you-first argument. I went first. Your turn.

J

The real line can't be constructed or really even imagined without limit points. Otherwise you get the Algebraic Numbers at best, that is the non-imaginary parts of the algebraic closure of the rationals. However that set is not the real numbers, is countable, and has measure zero. To get further you need limit points and pretending otherwise is silly.


The real numbers are not axiomatic or "irreducible", they are built from the set of integers in a very well defined and rigorous way.

 

[onejayhawk]

Except that you haven't since the bolded portion of your proof is false.

Consider the open set (0,1), since we are in a metric space this means that for every point x in (0,1) there exists an open ball centered at x wholly contained by (0,1).
Now suppose x=0.99... and x is in (0,1), then for some epsilon the open interval (x-epsilon,x+epsilon) is wholly contained in (0,1), however 0.99...+epsilon > 1.
Therefore either (0,1) isn't an open set (it is) or 0.999... isn't contained in it.

Nope. 0.999... is an element of the open segment (here is the best part) by definition. The design of the series confines it within the open segment. Every iteration of the series moves it further to the right, but still in the segment. Infinite iterations move it infinitely close to the (open) endpoint. They cannot move it past.

Again with the epsilons. I though we were past that particular gimmick.

The real line can't be constructed or really even imagined without limit points. Otherwise you get the Algebraic Numbers at best, that is the non-imaginary parts of the algebraic closure of the rationals. However that set is not the real numbers, is countable, and has measure zero. To get further you need limit points and pretending otherwise is silly.

I might grant you that much.

The real numbers are not axiomatic or "irreducible", they are built from the set of integers in a very well defined and rigorous way.

Of course they are. Can you define either "1" or "+"? Without that much you cannot even construct the integers. Transcendental numbers are neither more or less imaginary than i.

J

 

[warpus]

So. You're saying that arrows never reach their targets?

Some of them indeed don't, due to bad aim.

Let's say you're firing a virtual arrow from the number 0 to the number 1. The only way for you to reach your target is to first mark the half-way point, write it down (0.5), and then write down the half-way point between where you end up and the target (0.75)

You repeat this exercise until you write down the number 1.

You will never write down the number 1. And so the virtual arrow will never reach its target, because you'll be stuck dividing things in half forever.

Real arrows of course reach their targets, if the aim isn't off and the wind and other factors are favourable. But real arrows aren't tasked with the division of distances into halves for all eternity, all they do is obey the laws of physics.

 

[HorseshoeHermit]

Nope. 0.999... is an element of the open segment (here is the best part) by definition. The design of the series confines it within the open segment. Every iteration of the series moves it further to the right, but still in the segment. Infinite iterations move it infinitely close to the (open) endpoint. They cannot move it past.

1. Why can't infinite iterations move it past?

2. They don't have to move it past, they only have to move it as far as that endpoint.

3. My previous questions.

 

[nc-1701]

Nope. 0.999... is an element of the open segment (here is the best part) by definition. The design of the series confines it within the open segment. Every iteration of the series moves it further to the right, but still in the segment. Infinite iterations move it infinitely close to the (open) endpoint. They cannot move it past.

0.99... is not a member of the set of expansions 0.9, 0.99, 0.999, ...

That is a fundamental misunderstanding which you apparently cannot move past.

 

[Kyriakos]

Some of them indeed don't, due to bad aim.

Let's say you're firing a virtual arrow from the number 0 to the number 1. The only way for you to reach your target is to first mark the half-way point, write it down (0.5), and then write down the half-way point between where you end up and the target (0.75)

You repeat this exercise until you write down the number 1.

You will never write down the number 1. And so the virtual arrow will never reach its target, because you'll be stuck dividing things in half forever.

Real arrows of course reach their targets, if the aim isn't off and the wind and other factors are favourable. But real arrows aren't tasked with the division of distances into halves for all eternity, all they do is obey the laws of physics.

Real arrows *may* do that, but surely we do not observe them do it
In Amstrad games of old i recall being interested in what happened when the disc was malfunctioning, and on the screen you would get weird corruptions in the game, for example a floor there in the actual game would now not be there, or be re-appearing as many times as you jumped from it (interesting, no? ). One has to assume that the program was 'trying' to follow through other (non-corrupt) parts, while carrying with it a corruption as well. A bit like carrying a sensory stimulus that is not tied to 'reality'.