# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

Math is logic, but it must start with some assume or defined objects. A "point" is one of these that by definition (an axiom) has no size. Thus if you have a line segment between point A and B which are only 1 micron apart, there are an infinite number of point, all different form each other on that 1 micron line segment. Even more startling is fact that the number of points between A' & B' is an infinity just as large as if A' & B' were a trillion light years apart. You "common sense" is wrong as this can be proven. Basically this proof goes like this. given any point on AB, P, there is one and only one point. P',on A'B' that correspond to P (and conversely). For a specific example, consider point x which is closer to A than to B by the fraction 41/ 333 or 0.123123123123123123.... there is the point x' (the only one) that is also 0.123123123123123123.... closer to A' than to B'.

As far as to which of the 1/3 slices of pie the center point "belongs to" the choice is arbitrary. If initially it belongs to piece #2 and someone say: No, it should belong to pieces #1, then I say OK. move it. Do you think after then move that the area of piece #2 has decreased and that of piece #1 has increased?

If yes, it just shows you are illogical, of have forgotten the axiom defining what points are.
If I remember correctly we had this discussion before concerning Pi.
the result of this is to refer to boundary (topology)
http://en.wikipedia.org/wiki/Boundary_(topology)
which resolves the issue apparently.

And it is similar to the issue of reducing a spheres diameter by infinity - what do you actually end up with? [sort of thingo]
zero or non-existence

0.77(7)

which is 0.777777778
How do you figure that?

Obvious question is why the above and not
say, 0.7777777777778 ?
or 0.77778?
or 0.777777777777777777777777777778
There's no 8 anywhere in 0.(7)

How do you figure that?

There's no 8 anywhere in 0.(7)
hmmm calculator seems to perform the same as it does with 0.99(9)
7/9 = 0.777777778 according to my calculator
and 0.7(7)
also = 0.777777778

7/9 is exactly equal to 0.(7), but rounded to 0.7777778 (or however many digits on the display) on your calculator.
9/9 is exactly equal to 0.(9), and exactly equal to 1. The calculator of course displays the simpler representation.

If I remember correctly we had this discussion before concerning Pi. ... it is similar to the issue of reducing a spheres diameter by infinity - what do you actually end up with? [sort of thing] zero or non-existence
No. comparing the number of points on 1 micron long line segment to number on 1 light year long line segment (or inside a cube one light year on edge) is not related to value of pi problems or topology problems. It is more related to "Measure Problems."
The answer to your question is you end up with a sphere of zero radius. You can't change dogs into ants either by breeding them ever smaller - to be smaller than most ants - they are still dogs.

I probably should leave it at that, but note the zero radius sphere has a zero area of surface with an infinite number of points on it as even the sum of an infinite number of points has zero area or is equal to 0.0000000... in area.

... {my} calculator seems to perform the same as it does with 0.99(9)
7/9 = 0.777777778 according to my calculator and 0.7(7) also = 0.777777778
After you did not pay up and never agreed with any of my several proofs that something, normally called a photon, does exist between A & B as light travels from A to B, I have thought you were a "cheap guy" - next time buy the infinite calculator.

hmmm calculator seems to perform the same as it does with 0.99(9)
7/9 = 0.777777778 according to my calculator
and 0.7(7)
also = 0.777777778
Your calculator rounds. Try pen and paper.

You do remember how to do division, don't you?

Your calculator rounds. Try pen and paper.

You do remember how to do division, don't you?

No, he doesn't. He never did. This is why he keeps trolling.

I probably should leave it at that, but note the zero radius sphere has a zero area of surface with an infinite number of points on it as even the sum of an infinite number of points has zero area or is equal to 0.0000000... in area.

That's utter nonsense. A sphere is a set of points equidistant from the center. If the radius is zero, a sphere is the set of points equidistant from the center. It consists of exactly one point, the center.

In Euclidean 3-space, what is the sphere of radius zero about the origin? By definition it's the set of points at a distance of zero from the origin. In other words it's the set consisting of the origin. One single, solitary, identifiable point.

... In Euclidean 3-space, what is the sphere of radius zero about the origin? By definition it's the set of points at a distance of zero from the origin. In other words it's the set consisting of the origin. One single, solitary, identifiable point.
I agree all the point on the surface of the sphere with zero radius are at the same location as the point at the center of the sphere, but they still have their defined identity. Just because two definition both refer to the same thing (or set of things) does not mean the definitions are the same.

The most famous example of this in the "real world" not math only, was there were two named stars: The evening star and the morning star. After a few dozen centuries, the more educated knew that they were both Venus, but just like the zero radius sphere, this did not destroy how the were defined or the fact their definitions were different. I.e. the surface points of the sphere are, as you said, are defined by being equally distant from the central point and in the zero radius sphere case they are at the same point as the central point is, In math, especially, you must use the definition of the set to say things about it.

For example it is true that the evening star has inspired many more poems of love than the morning star ever has. etc.
What is "non-sense" is to say: "Venus has inspired more poems of love than Venus has." Ergo: Definitions are important when making statements about the defined objects.

But even as I made that final note, I knew few would be logical enough to recognize the points of the sphere's surface are not defined by their location being the same as the center of the sphere in this special zero radius sphere case.

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I agree all the point on the surface of the sphere with zero radius are at the same location as the point at the center of the sphere, but they still have their defined identity.

So when I consider the point (0,0,0) in Euclidean 3-space, I should really consider it to be a huge collection of points that happen to be in the same place but that are individually distinct?

Pass the bong, dude! I like what you're smokin'.

Good morning, Billy T, everyone.

Math is logic, but it must start with some assumed or defined objects. A "point" is one of these that by definition (an axiom) has no size, but has a location. Thus if you have a line segment between point A and B which are only 1 micron apart, there are an infinite number of points, all different from each other by their locations, on that 1 micron long line segment. Even more startling is fact that the number of points between A' & B' is an infinity just as large (and not bigger) as between A & B, even if A' & B' were a trillion light years apart.*

You "common sense" that there must be more points on line segment A'B' than on the 1 micron long AB, is wrong as this can be easily proven. Basically this proof goes like this:
Given any point on AB, P, there is one and only one point, P',on A'B' that correspond exactly to P (and conversely). For a specific example, consider the point x on AB, which is closer to A than to B by the fraction 41/ 333 or 0.123123123123123123.... there is the point x' (the only one) that is also 0.123123123123123123.... closer to A' than to B'.

That's ok mate, I know all about that 'mathematical points on the number line' axiomatic logic flow results. No sweat.

I am exploring your and rpenner's 'takes' on the 'point' in the REALITY context of MD's example of a REAL PIE disc (and of my example of the Modulo 12 treatment of SIMULTANEOUS 'superposed' "0"-is-"12" on the face of the 12-HOUR Clock-----please read my posts #146 & #152). Thanks.

As far as to which of the 1/3 slices of pie the center point "belongs to" the choice is arbitrary. If initially, when I cut conceptually the pie, it belonged to piece #2 and someone say: "No, it should belong to pieces #1, as that was the first cut." Then I say OK, lets move it to be part of piece #1.

Do you think after the move that the area of piece #2 is smaller and that of piece #1 has increased? If yes, then you are illogical, or forgot the axiom defining what points are.

*likewise, there are no more points inside a cube one light year on an edge than on the 1 micon long line AB. Here is a nearly completed hint for you to proof that:
Consider the point inside the cube with Cartesian coordinates (a,b,c) and note that a, b, & c are expressed as decimal fractions of one light year. I.e a is a string of numbers 0 thru 9 like 0.2640684772456 without end usually but could be exactly 0.2 for the infinity of points in the bc plain with coordinates (0.2, b, c). In general a = a1 a2 a3 a4 etc., where for any point in the cube with a= 0.2640684772456 then for it, a1 =2, the a2 =6, the a3 = 4, the a4 =0 etc. for b & c.

Now on the 1 micron long line corresponding to point in the cube at (a,b,c) the is one, and only one, point with x-axis coordinates = 0.a1 b1 c1 a2 b2 c2 a3 b3 c3, a4 b4 .... etc.

And of course for any point on the 1 micron line there is one and only one corresponding point inside the light cube.

If there is an apple for every orange (one and only one corresponding to it) and an orange for every apple, then there are no more oranges than apples and no more apples than oranges - even your intuition should tell you that. But some infinites are bigger than others, but I won't detour more to prove that.

Again, mate, it's not the 'point' as JUST the math axiomatic entity PER SE that this PHILOSOPHICAL discussion is exploring, but that math-axiomatic 'point' relation to the REALITY of things which exist in the full reality range of existence from 0 to infinity, and what those two things called "0" and "Infinity" mean when examined in ALL contexts so that we can 'bridge' the gap between maths and reality better. That's where all my discussions are coming from.

No. comparing the number of points on 1 micron long line segment to number on 1 light year long line segment (or inside a cube one light year on edge) is not related to value of pi problems or topology problems. It is more related to "Measure Problems."
The answer to your question is you end up with a sphere of zero radius. You can't change dogs into ants either by breeding them ever smaller - to be smaller than most ants - they are still dogs.

I probably should leave it at that, but note the zero radius sphere has a zero area of surface with an infinite number of points on it as even the sum of an infinite number of points has zero area or is equal to 0.0000000... in area.

By that logic there in that last sentence (as I bolded above), you have just effectively said that:

- (mathematically) a point (zero) is Infinity (sum of infinite points is zero)? That is------"0" = "infinity"------and/or vice-versa?

And by that same logic, you would have the reverse to the summation process, ie DIVIDING a point infinitely many times:

- dividing (mathematically) a point (zero) infinitely many times would give a point (zero consisting of infinitely many points)? That is------"0" / "infinity" = "infinity" of points?------(or, again/else mathematically) an 'infinity of points' that is axiomatically STILL zero?

See that math-to-reality logic/relation gap I am exploring and trying to 'bridge', Billy T, everyone?

My discussions are all aimed at enhancing the maths-axiom-set so that it CAN then handle these 'gap' issues consistently with the reality from the very start, and so better reflect reality in its application, and minimize 'outputs' like "undefined" and "infinity" etc?

Please then, everyone, read this post (in conjunction with my posts #146 & # 152) carefully, so that there is no misunderstandings leading to kneejerk repetition of what we all already know. We are after NEW insights aiming to enhance the axiomatic set, not just rehashing old maths-only views based on current axiomatic set.

Thanks, Billy T, everyone.

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After you did not pay up and never agreed with any of my several proofs that something, normally called a photon, does exist between A & B as light travels from A to B, I have thought you were a "cheap guy" - next time buy the infinite calculator.

Ahh you are quite correct as usual... my smart phone rounds up but my business/financial calculator doesn't.......

normally called a photon, does exist between A & B as light travels from A to B,
1] and note as you missed the point before that the proof must prove the photons "independent existence" from mass that may be reflecting/imitating it.
and btw the \$500 usd still stands [no takers yet]... [chuckle]
2] And I am tempted to add another prize offered to any one who can calculate to an appropriate approximation, the amount of EM energy, supposedly in transit universally at any given moment...and especially explain why this appears to be unaccounted for by mainstream science.
3] A consistent and non-ambiguous definition of Energy would also be nice...

That's utter nonsense. A sphere is a set of points equidistant from the center. If the radius is zero, a sphere is the set of points equidistant from the center. It consists of exactly one point, the center.

In Euclidean 3-space, what is the sphere of radius zero about the origin? By definition it's the set of points at a distance of zero from the origin. In other words it's the set consisting of the origin. One single, solitary, identifiable point.

I agree , [for what it's worth] however there is more to it if you delve more deeply I think...

Imagine you are out in the void of interstellar space... and you wish to consider a location. a set of co-ordinates.
You reduce a sphere infinitely to determine a point in that void.
I would guess that you would end up with a material location yet also an immaterial non-existeieieie*nt sphere.
It is similar I feel to what Heisenberg was exploring with related outcome being the uncertainty principle. - a paradox exists and we logical thinking entities just wont accept it.
This paradox [ which I can prove easily and empirically: re: Attraction paradox] then filters through all our Physics, Math, philosophy etc without being properly addressed IMO

"Achilles can NEVER beat the Tortoise to his position in the race...no matter how fast he is going"

*oh no ... a "Tach Event" is occurring ... beep! beep! [chuckle]

@undefined
By that logic there in that last sentence (as I bolded above), you have just effectively said that (mathematically) a point (zero) is Infinity (sum of infinite points is zero? That is,------"0" = "infinity"------and vice-versa?.

it certainly looks like it... and if so leads to these types of discussions endlessly...
well math does terminate infinity using limts after all...

"The quantification of that which is unquantifiable"

@undefined

it certainly looks like it... and if so leads to these types of discussions endlessly...
well math does terminate infinity using limts after all...

Hence my approach to these discussions with the aim of trying to end same once and for all by using the contextual reality approach to enhancing the maths axiom set and making maths more consistent to reality right from the starting logics, rather than having to keep 'fixing' it with 'loophole' statements like "undefined" and "limits" and "infinities" etc.

The discourse continues (hopefully calmly and politely without all the emotionally charged 'passionate attachment' to status quo just for the sake of it).

I observe the discourse in the spirit of co-operation and change for the better of both the maths and the physics logics/reality 'starting point' (pun intended ).

Thanks for all the interesting discussion. Please carry on, QQ, everyone!

Hence my approach to these discussions with the aim of trying to end same once and for all by using the contextual reality approach to enhancing the maths axiom set and making maths more consistent to reality right from the starting logics, rather than having to keep 'fixing' it with 'loophole' statements like "undefined" and "limits" and "infinities" etc.

The discourse continues (hopefully calmly and politely without all the emotionally charged 'passionate attachment' to status quo just for the sake of it).

I observe the discourse in the spirit of co-operation and change for the better of both the maths and the physics logics/reality 'starting point' (pun intended ).

Thanks for all the interesting discussion. Please carry on, QQ, everyone!
Whilst your idealism for grounding math in the real world is admirable you may also have to consider grounding the call for change in the real world as well.
In physics things can change rather easily compared to Math. IMO To instigate a global axiomatic change in the field of mathematics even if agreed to and wanted would take many many years of politics, discussion, theorizing.
But no doubt you will say, "Gotta start somewhere mate!" Sheesh! , Zeno of Elea wrote his paradoxes thousands of years ago and we are still struggling...

The most famous example of this in the "real world" not math only, was there were two named stars: The evening star and the morning star.

So are you saying that there are a lot of NAMES for (0,0,0)? Or that there are a lot of distinct points that happen to be in the same location?

Surely there are many names for things. In fact even in Euclidean 3-space we can choose a different basis and thereby have a different name for (0,0,0). But it's still the same point.

Your example is extremely weak, using the fact that there are many names for the same object, to argue that there are millions of points all residing at a single point of Euclidean 3-space.

Whilst your idealism for grounding math in the real world is admirable you may also have to consider grounding the call for change in the real world as well.
In physics things can change rather easily compared to Math. IMO To instigate a global axiomatic change in the field of mathematics even if agreed to and wanted would take many many years of politics, discussion, theorizing.
But no doubt you will say, "Gotta start somewhere mate!" Sheesh! , Zeno of Elea wrote his paradoxes thousands of years ago and we are still struggling...

Twas ever thus, mate! However, you forget we now have the INTERNET. Global communications and information systems/dynamics which the ancient "Minds" and "Schools" pondering Natural Philosophy would have made great use of!

It's intriguing to ponder just what ELSE those ancient minds/philosophers MIGHT have come up with THEN if they TOO had the global/instant communications/information access WE LUCKY modern minds have now?

Things evolve faster nowadays because of the internet and globalization, QQ; that includes stodgy old mathematicians and physicists (if there ARE any such still extant after the internet age fell upon them and the rest of us! ).

I proceed as indicated by the need for 'bridging' the maths-reality gaps as best and as quickly as can be done by whatever approach works best. One can do no more, and certainly no less if one has the best interests of humanity, science, maths and physics reality at heart and in mind, hey QQ?

Cheers and good luck and good thinking and good discourse to you, QQ, everyone!

PS: QQ, I have to log out soon for a few hours (or maybe until tomorrow). So see y'all when I get back! Cheers.

Part of this problem of reality vs abstraction is that the use of infinity applies to that which exists or is existent.
When we talk of 0.999... = 1 we are simply confirming that infinity does indeed lead to a existent outcome. [but only because of the use of an arbitrary limit]

Alternatively the use of the infinitesimal as a concluding measure of infinite reduction is also a pseudo material value.

Infinity applies to that which is "something" and using an infinitesimal is an abstract notion of the border or boundary between "something" and "nothing"
If we reduce a sphere infinitely and place no limit or termination upon it we have the situation of an unquantifiable "something-ness" hence placing a boundary or limit, grants us quantifiability. [human nature 101]
So philosophically therefore, one can conclude that if you reduce a sphere infinitely the volume of that sphere must be non-existent. [ not zero - as all values vanish completely and zero is currently used as a relative value and not a non-value]

However because we are left with a shell that has zero radius, yet has a volume of nothingness, the "point" exists as a zero point that happens to have a volume of non-existent space.
If we retain the fact that it is a zero dimensional sphere then it must have volume with in it.
this is the nature of the paradox.
"A zero point sphere with a volume of nothingness" thus proving the existence of "nothing" by default of infinite reduction. [using a 3 dim sphere to show the case]

If in doubt ask the following question:

If the 3 dim. sphere has a zero radius and considered to be still existent as a point, does it still have volume with in it?

The answer can only be logically that for it to be a sphere It must have volume. yet that volume is non-existent. [paradox]

"The resultant sphere is 3 dimensional with a zero dimensional volume." [paradox]

Now if accepted , even tentatively, that void of nothingness [zero dimensionality] is the anti thesis or the opposite of everything that is existent [3dim], how influential is that void?
And note that all objects of mass can be reduced "virtually" to their zero point thus identifying the location but not the "something" of the center of mass [ center of gravity]

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