An Infinity of Infinities

A known distance divided by a known interval will return a known number of points.

Reducing the interval will never make the number of points infinite.

(although your calculator may indicate so)

You may be confused with the half of the remainder moves.

That is a dtudent....I mean a student I remember when I was in college, my professor was talking about hyperbolas or something. And he said something similar.

(2) people facing each other, (2) steps away...about 5 feet. And one of the 2 people talks a step toward the other, cutting the distance in half...30 inches. He takes another step, half that (15 inches) and another step 7.5 inches. his step is always half the distance, so hypothetically the person will never reach his destination,even thou its a finite distance. because his steps are always half of the required distance........so.......smole em if you got em.

"smole em if you got em."

Have you been smolen already?

That was some infinity time ago .

I remember a professor giving the same example, except he identified the two people as a man and a woman. He observed that if you posed the question, "Will they ever meet?", to a mathematician and an engineer, you would receive two different answers. The mathematician will immediately and with certainty, reply, "No, they will never meet!" The engineer will hesitate, then say, "Well, they will get close enough for all practical purposes."

The theoretical and the practical, but what happens when you insert Heisenberg into the equation?

Good Point of view

Suppose the known distance is 2" and the known interval is 0"?

Zero is not infinitesimally small !!!

0.000000000000000000000001 is so is 0.0000000000000000000000000000000000001

With zero all your lines will be in one place (touching each other - as stated in the OP)

Hendrik wrote: Zero is not infinitesimally small !!!

You're right!

What have you been smoking? How can you put an infinite number of lines and spaces in a finite distance?

lynlynch wrote: "How can you put an infinite number of lines and spaces in a finite distance?"

That's the same as considering the real number line--between any two real numbers you can always find another real number. That is an infinity. ;-)

OK.

Boring.

Here's a 2-dimensional viewpoint, in color, zooming. (Not mine.) There are lots of these out there:

fractals

Ok,let me rephrase the question:Take 2 concentric circles.Put an infinite number of lines from the center of the smaller circle to the edge.There is an infinitely small space between the lines.Extend the lines into the outer circle,and they will diverge,yet there will still be an infinite number of lines, yet the spaces between them grow larger as they grow longer and further from center.Eventually,the spaces between the lines will approach infinity.Hence all infinities are not equal,and are only possible within a Finite context, and very possibly only within our mental construct that we perceive as "reality",although ordinary matter makes up only a very small percentage of the universe.We are like smoke,and are merely ghosts or shadows of true reality.

Pass the wacky 'backy and please,don't Bogart that joint.

MellowYellow

There is an infinitely small space between the lines.

and

Hence all infinities are not equal,

its pretty hard to put a value on infinity....^{death to the infidites.}

^{before long you'll be talking about string theory}

Corollary of "all infinities are not equal":

- an infinity of bottles of wine

- an infinity of elephants

Both are abstract concepts and they'll both get you smashed.

So, in consequence, these two particular infinities are equal. Albeit slightly different. Reductio ad absurbum methinks.

Anyway, it would be best if the OP looked up the mathematical concepts of Limit Theory where a proposition can be seen to tend towards a result, used extensively in convergence of series.

That is the difference between mathematicians and engineers. Mathematicians, particularly set theorists, distinguish between "different" infinities ever day when they discuss the "degrees of infinity" or cardinality of a set of numbers.

The easiest way to think about it is there are two types of infinities, countable and uncountable. The countable infinite sets allow you to "map" or pair every number with a number in the set of natural numbers or the counting set (1,2,3,4,...). As an engineering example, any level of precision will produce an infinite number of measurements (1mm,2mm,3mm,4mm,...) or (0.1mm,0.2mm,0.3mm,0.4mm,...). Another example is Zeno's Paradox of walking to a wall and halving the distance. Again this is countable (1/2,2/3,3/4,4/5...).

It was in the late 1800's that mathematicians began to look at "uncountable" sets and distinguish them from those sets above. The set of all Real numbers between 0 and 1 is an example of an infinite, but uncountable set. That is to say that you cannot establish a method that would allow you to count all possible numbers between 0 and 1 unlike the examples above.

Ok so you now wish to explore the set theory ideas of Georg Cantor. In his work, Cantor proposed that there are infinities of differing levels of complexity. I've personally found these concepts fascinating but of little real use. I only dabbled in this mathematics at the undergraduate level many years ago. It is true that mathematics doesn't have to have a real use to be worthy of an in depth study. Maybe this mathematics can eventually reveal a hidden fold of reality.

Thanks for the link,it led me to this, which I think is very appropriate to this discussion:

William Blake - Auguries of Innocence

To see a world in a grain of sand, And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour.

Infinitesimal quantity is a relative concept as Infinity.

When you divide the small circle into infinitesimal sectors, and extend to the bigger circle, then you must also grow your perspective.

Fractals... as you go from very near to very far, you pass by different views of the picture even though the initial pattern is the same but viewed from different distances...

Since the Universe is infinitely big (Theoretical ), then you go back infinitely far to see that your expanded sector edge gets smaller depending on the position of the viewer...

maybe not very clear but since infinity is involved, infinitely big and infinitely small gets to be dealt with simultaneously...

To be able to see the infinesimally small, you must reduce yourself to that level and then it is not so small again!

Anyway, these are theoretical concept to aid reasoning in the philosophical mathemetics (Does it mean anything?..)

"To be able to see the infinesimally small, you must reduce yourself to that level ..."

So you reduce yourself to the level you had once thought was infinitesimally small, and find you have an infinite set of reductions yet to go. Now what? Incidentally, how could you "measure" how small you've already gotten; how much smaller you have yet to get? Without measure, how could you reverse course, add all that back up, and determine the length of your journey?

I always thought the best handle on the concept of infinity was considering it either as "the place" where all parallel lines meet, or as a mathematical construct which allows inquisitive minds to put off having to define the opposite of "finite" - i.e. that which is NOT countable.

Now that we have governments that do that for us, by generating ever increasing debts to satisfy our ever-increasing lists of entitlements, and don't need the other constructs anymore.

COUNTABLE. Thus, OP's original post is flawed in it's presumption of countable measure to measure uncountable actions.

I'm sorry, but that is confused and incorrect. The natural numbers are infinite but countable, as are the rational and algebraic numbers. These sets all share the same cardinality, aleph₀. (Sorry, the Hebrew alphabet doesn't show up on the menu.)

Excuse me but I thought that the rational number set was not countable. Certainly one can count the infinite set of thirds but this is just the easily counted set of integers divided by three. When the domain constraint is the set of rational numbers one does not know what comes after one third to make a count. 4/9 is after 1/3 but 10/27 is before 4/9 and 0.333333333333334 is between 1/3 and 10/27 too. So one is not certain which of these numbers should be counted next. All of these numbers are clearly rational (can be perfectly defined by a ratio). When one sets the numerator or the denominator to a fixed scaler value or even some functional relationship between these two sub-functions then one defines a countable set. A defined set of the rational number set is countable but I thought the entire rational number set was not countable.

I have the same argument for the algebraic number set for it includes the imaginary number set. Some countable algebraic number sets therefore have multiple entries for each count but a countable sequence does exist.

We're diverging greatly here from the original posting but it sure is fun.

Go to the lecture series referenced by Usbport (Post #25). In Part 5, "The Smallest Infinity", starting about five minutes into the lecture, Cantor's method of establishing the countability of the set of all fractions is explained. Maybe another lecture in the series deals with the algebraic number set. I haven't watched them all.

I apologize for any confusion I might have caused you. Please point out where that occurred, and I will do my best to be more clear.

As for lack of correctness - without using terms YOU, in your response, not I, in my post, introduced - please direct me to where it occurs and I will consider your point/s.

Otherwise, I stand by the OnThread nature of my post and await your legitimate challenge thereto.

(As to the menu's lack of Hebrew alphabet functionality, I suspect there might be an infinite number of alphabets yet to be represented on the menu; sadly, that might be another thread's topic...)

The most troublesome statement was this:
"...or as a mathematical construct which allows inquisitive minds to put off having to define the opposite of "finite" - i.e. that which is NOT countable."

As had already been discussed, the rational number are countable, because they can be placed into one-to-one correspondence with the natural numbers (positive integers). So "infinite" is not necessarily equivalent to "uncountable."

Of course, all this assumes we are speaking mathematically, not merely folklorically.

So "infinite" is not necessarily equivalent to "uncountable."

Okay, considering infinite as not necessarily equivalent to uncountable, then (if you will?) momentarily dismissing your phrase "not necessarily" - please reconsider my post regarding "how do I measure" without resorting to additional terminology which may or may not be relevant, such as "rational" or "natural" numbers.

This is, after all, infinity we are discussing: I get half-way there, report back my progress (assuming MEASUREABLE quantities are so far involved), then I sigh and volunteer (have I a choice?) to continue my folkloric journey into infinitesimalism whilst wondering "how much further must I divide myself to prove that I'll NEVER be able to divide a finite MEASUREABLE entity (my travel distance so far) into an infinite number of increments separated infinitesimally (the gaps into which I have yet to travel)?"

NEVER, of course, meaning "not merely folklorically" but also meaning both an infinite amount of negative time and the sum of infinitesimally small amounts of positive time. Whew; what an envelope of +/-.

Georg was truly enlightened; he didn't provably conceive the one-ended stick, but he bravely presented a conjecture using it. Could we not extend it (pun intended)? What would be the surface described by laying all such single-ended sticks end-to-end? IN HOW MANY DIMENSIONS WOULD SUCH A THEORETICAL SURFACE LIE?

I'm sorry it that last paragraph was unclear. I'm even more sorry if it still does not support, in your opinion, my contention that OP was in error by his supposition that an infinite number of infinitesimal divisions could, in fact, be quantified; be measureable - no, be PROVEN - to be OTHER THAN FOLKLORICAL?

I regret my "political" aside in my original post, one that might have affected your response. Folklore, as well as mathematics in the shadow of infinity (poetic license presumed), is relative.

It is MEASURE which rules our current understanding, and OP's conjecture lacks same. Or so I contend.

Regards,

Gene

I'm sorry, but all of that was just blather, and not very clear. Please go take a math class in the subject.

It is easy to find an infinitude of positive quantities that add up to a finite amount; e.g. Σ_n=1^∞ 1/2ⁿ = 1. But Σ_n=1^∞ 1/n = ∞. (The typography is limited here, but still good enough.)

Okay.

So, what is it about my "blather" that concerns you? Comprehension? On whose part?

Something, it must be, that allows you to feel safe with a recommendation that "a math class" might be in order for the "offender" against your presumptions; something that your symbology, somewhat akin to the paintings of cave dwellers so many years ago, might dispell by their very submission?

I proposed thought questions, you provided rote, and then, well, squiggles of symbology. That would be "blather" to me, were I not so overwhelming curious and so inadequately concerned with insult.

Sad to say, I have not revisited this thread until now.

Your reasoned and - forbid me saying this if you must - enlightened response is elicited. Which "math class" might you recommend? lmgtfy (no caps intended) might just tickle your fancy, then again it might just have led to your initial response: I'd ask that YOUR answer not be something as mundane as "because THEY said so..." or nothing at all.

Either way, I apologize sincerely to you for the lack of clarity that you sensed; I have challenged you to go one step further, if you can, to offer clarity where you sense it missing.

Until that happens, I must simply apologize also for the time lag in responding to your non-specific and flippant response.

Might I suggest this to you, having not yet undertaken your "math class": Inadequate typography being only one possible issue; ease of finding an infinitude of positive quantities adding up to ANY finite amount - SHORT OF TYPOGRAPHY - might be the KEY issue. Cave art aside, I mean.

Regards,

Gene.

I don't know if second-year algebra these days covers the same material that it did in my time (1964-5), but that is where I first encountered this. The concepts are very well known, and not very complicated. However, you have failed completely to use these mathematical terms in the ways they are commonly understood. To put it bluntly, you have no idea whatsover of what you are trying to talk about, and you (not I) are completely confused. Moreover, your language style is pretentious, irrelevant, and offensive.

I think that this is an ancient conundrum called "Zeno's Paradox." The ancient Greek philosopher Zeno posed this one - being unfamiliar with the concept of a converging series.

Couldn't figure out how to insert the Wiki link.

What you must never forget is that infinity is only an abstract for something beyond currently measurable quantities, or well beyond required measurement. - A box of 1000 bolts could be considered an infinite supply, if only one was needed each year. To infinity - and beyond!

This is very easy case. Let's use this method of division: divide your distance L into 2 halves. Then you have 2^1 pieces L/2^1 length each. Then divide each piece in half. You will have 2^2 pieces L/2^2 length each. Etc. Doing it n times you get 2^n pieces L/2^n length each. However, the total length is 2^n*L/2^n = L. You can now divide it infinite number of times (taking n -> inf) the summed length of infinite number of infinitesimally small pieces will be L. That's all into it.

Degrees of Infinity:

http://wn.com/aleph_number

Here is a screenshot from the link. (NOT me in the video.)

Whatever you're drinking, can I have a pint too, please?

Precisely Cantor's work. GA

All this is clear, but like those nineteenth century respondents to Cantor's insight, I still find it "unsettling" that infinities can be of different sizes, and that one infinity can contain another. In high school math class this concept was illustrated with a Venn diagram. Simply draw a circle (which will contain an infinite number of points), then draw another circle around the first circle. The larger circle will contain and share the smaller circle's infinity of points, and in addition will also contain an infinity of unshared points.

Outside of math class though, infinity remains a difficult concept. I have never found a stick with just one end. But according to Cantor (and I accept this) if such one ended sticks did exist, they could still vary in length. I've known this for a long time but it still seems like something out of Alice in Wonderland.

I see your "point", but suppose the small circle represents the set of all integers 1,2,3..., and the larger circle (which contains the small circle) the set of real numbers? Didn't Cantor demonstrate that since a one-to-one correspondence could not be established between these two sets, they belong to different orders of infinity?

If the lines you describe connecting pairs of points in the two circles were parallel lines then yes, there would be a one-to-one correspondence, but I would argue that if it is necessary to draw non-parallel lines to connect the pairs then it is no longer one-to-one because each connection is angularly unique.

Both circles would have to be the same size for the one-to-one correspondence to exist. If the circles are different sizes, they belong to different orders of infinity. For example, the circle representing the set of integers 1,2,3..., and the circle representing the set of even integers 2,4,6..., would be concentric and the same size, because of their one-to-one correspondence, even though common sense tells us that the number of all integers should be twice the number of even integers. All the lines connecting the pairs of points in these two sets would be parallel.

I agree that it's mind boggling, but we are in good company. I read that Georg Cantor died in a mental hospital in 1918.

Sorry, but virtually everything here is quite incorrect. The small circle cannot be put into one-to-one correspondence with the integers. Cantor demonstrated something else: that the rational numbers could not be placed one/one with the real numbers.

One-to-one mappings do not require parallel lines. The phrase "angularly unique" is irrelevant (or improperly applied) in this context.

Etc.

The example I provided of the equivalence of the infinity of all integers with the infinity of even integers is taken from Isaac Asimov's "Biographical Encyclopedia of Science and Technology" in the entry under Georg Cantor.

As I understand Cantor (and Asimov), circles representing these two sets would be concentric and equal in size, whereas a circle representing the set of real numbers belong to a higher order of infinity and would contain any set of integers (which could be represented by a smaller circle(s). There would be no one-to-one correspondence between all of the points contained in the larger circle and those contained in any smaller circle. When a circle is drawn such that it contains another, by definition more than one order of infinity has been illustrated, and one to one mapping cannot occur. Such one to one mapping could only occur if the sets belonged to the same order of infinity. Those sets would be represented by equal size circles.

If we accept that any circle can be mapped one-to-one with any other circle, regardless of size, are we not then reverting to the pre-Cantor belief that all infinities are equal? Surely there must be some consistency and rigour in how that mapping is defined. Thus my reference to parallel and non-parallel lines.

I'm sorry again, but that's even worse. Neither of those sets of integers can be represented correctly by a circle, arc, line, or line segment. I can't imagine that Asimov or Cantor ever said such a thing.

This is exactly the nuance complications that one finds in discussing Cantor's set theory ideas of infinities. I remember but cannot get my hands on Asimov's explanation of Cantor's ideas. I think you missed a critical nuance.

One can certainly map an identical number of points on any circle onto any other circle. The simple proof of this is that when one converts the point mapping system to a circular coordinate system, there is one unique angle coordinate for each circle point. However, when one has two concentric circles one circle must reside inside the other one. So when the circles are the boundary that defines a set of points and one circle defining set is a subset of the other circle defining set then one infinity will be larger than another. For the larger set will contain all of the points of the subset and a defined infinite region larger.

Sorry, that last paragraph is incorrect. The two circular disks have the same cardinality, usually designated c (for continuum, which is also the same for the line, the plane, the real numbers, etc. Some of the mappings are tricker to construct than others.

Tornado is probably correct that Cantor didn't use Venn diagram circles to develop his ideas, nor in the example given did Asimov use them to explain those ideas. But the use of a Venn diagram is useful in visualizing Cantor's orders of infinity as they relate to various number sets. As I wrote, I remember seeing such a representation in an old math text. I don't think it was ever intended to be mathematically rigorous.

But just as a circle can be drawn to represent the set of all dogs that have fleas, so too circles can be drawn to represent the set of even integers, the set of rational numbers, the set of real numbers... Then, those circles can be placed in relation to each other - intersecting, overlapping, containing each other - in accordance with Cantor's insights. I acknowledge the mapping problems, but those problems aside, it's a useful and simple visual aid to understanding Cantor's work.

Infinity is a slippery fish. Until Cantor, it was "the big one that got away", but Cantor managed to hook and net this ultimate "Walter". For this he was attacked and ridiculed. Like the Old Man and the Sea, he "went out too far". It drove him mad in the end.

I agree that probably Tornado is correct about the cardinality of the different sets. I still believe that its a good pictorial to show that one infinite set can be larger than another. Every time I take a few minutes to glance at some web references on Cantor's work my eyes and brain starts to glaze over.

I wonder if any scientist or other skill set has ever found a use for Cantor's work. Maybe that could be another thread if anyone knows how to use these concepts.

_{Ahem Tornado, Jorrie?}

Number theory was once considered the queen of mathematics because it had no applications. That's not really true, of course, but it's the thought that counts.

Other than getting abstract concepts to be consistent, I'm not sure that infinity has any connection with the physical universe.

[For the earlier example of one circle twice the diameter of the other, here is a one-one mapping. For any point at polar coordinates (θ, r) in the first, match it to the unique point (θ, 2r) in the second. Going the other way, for any point (θ, r') in the larger circle, map to (θ, r'/2) in the smaller.]

Wacked ... infinitely smaller, added to infinitely smaller, is infinitely smaller, not infinitely larger in size, just infinite in number ... like the number of times we try to explain what God is vs what God is.

The number of small spaces increases to infinity... the sum of that space decreases approaching zero or should I have said that backwards ...

You can map out all the positive rational numbers via this scheme, like a spreadsheet:

Across the top list the natural number 1, 2, 3,..., to be used as numerators.
Down the left list them likewise, to be used as denominators.

You can easily find any positive rational number in the resulting table.

Now, starting like 1/1, 2/1, 1/2, 1/3, 2/2, 3/1, etc., you can count off the positive rationals by shuttling diagonally through the table. The table contains duplicates such as 1/2 and 2/4, but if the extras are countable, then all the more so are the "fewer" unique ones.

Yep. Whatever you've been drinking, can I have a pint, please?

Not quite. You can split a finite distance into infinitely many parts and it still stays finite cause each part gets smaller. Infinity doesn’t automatically mean something becomes infinite, it depends on how it adds up.