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

As an aside:
We have here a lane way dining area in the city center that is constant target of a group of organized street beggars. [In Australia social welfare is normally just adequate to prevent serious begging] Well this one particular Heroin Addict, worked to the philosophy that if he didn't have any money he was sure as hell going to make sure someone pays. So he would go into the restaurant areas in his drug induced stupor, terrifying everyone with his shocking and unpredictable state and sit down at any empty seat he could find,introducing himself to the table guests and wait till they paid him to leave.
Sad but true... it only ended after a quite few times when the police were called and eventually banned him from the area.
..and if you don't think oxygen and water are money, just wait till they finish destroying the Amazonian rain forests, and climate change and population growth impacts on our drinking water...

 

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Happy New Year 2014.

Consider A and B as two distinct points anywhere in the space so that a line segment A-B has a finite length. Denote the mid-point of this line segment A-B as B(0).

Consider the line segment A-B(0) and denote its mid-point as B(1).

Consider the line segment A-B(1) and denote its mid-point as B(2).

Continue these operations repeatedly(infinite times) so that subsequent mid-points will shift closer to the point A.

Let us denote the mid-point in general term as B(n) where 'n' is a non-negative integer from '0' to 'infinity'. So, when n is 0, B(n) is B(0); when n is 1, B(n) is B(1); when n is 2, B(n) is B(2); ... when n is infinity, B(n) is B(infinity).

If we compare the point A with point B(infinity), there will be a non-zero infinitesimal distance between these two points.Within this limit of non-zero infinitesimal distance between point A and point B(infinity), the consecutive point for point A can be placed.

It is believed(though wrongly believed) that a line is made up of points. These points cannot be placed side-by-side touching each other to make a line. These points have to be placed discretely to make a line. Thus the points in a continuum of a line are infinitesimally discrete.

 

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Happy New Year

Consider the decimal number, S = 0.000...(n-1) times 0 then 1; where n is a non-zero positive integer from 1 to infinity.

So that when,
n=1, S=0.1; here 1 is at the first digit after the decimal point.

n=2, S=0.01; here 1 is at the 2nd digit after the decimal point.

n=3, S=0.001; here 1 is at the 3rd digit after the decimal point.

n=4, S=0.0001; here 1 is at the 4th digit after the decimal point.
.
.
.

Similarly, when n=infinity, S=0.000...1; here 1 is at the infinity-th digit after the decimal point.

 

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Happy New Year.

A point does exist, though it is dimension-less or has a zero radius.

Without a point you can not locate a position in 3D space in terms of (x,y,z).

Without points you can not measure a distance in the 3D space.

Without points how can you locate a position in the 3D space?

 

All good up to these points that both say infinity not only is a number, it is an integer. This is untrue and therefore your demonstration falls apart.
Also you later use the word "limit" in a way unrecognized in English or mathematics.
The correct way to express the distance between A and B(n) is \(\bar{AB(n)} = \frac{1}{2^{n+1}} \, \bar{AB}\) so even if infinity was a number starting with a different initial length \(\bar{AB'} \neq \bar{AB}\) doesn't obviously lead to the same expression \(\frac{1}{2^{\infty+1}} \, \bar{AB'} =^? \frac{1}{2^{\infty+1}} \, \bar{AB}\) so the uniqueness of your alleged consecutive points is not demonstrated.
Finally, geometry will never allow you to prove the points are consecutive because

is a demonstration that there are no consecutive points in geometry.

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Points are not placed side-by-side to make a line -- that's the wrong concept because there are more than any finite number of points so no procedure exists to build a line by adding points one at a time.
But lines contain all possible points that are collinear with the defining points of a line. So when two non-parallel lines intersect, they intersect in exactly one unique point that is shared by both lines.
You allegation that "points have to be placed discretely to make a line" is baseless in geometry or set theory. You phrase "infinitesimally discrete" is meaningless in English or mathematics.

Again you think you are finding fault with beliefs, but what you are actually trying to do is argue with a definition which is absurd. Definitions are fixed beginnings of arguments, not items up for debate. If you want to start with new definitions, then you carry the burden of demonstrating that you are doing mathematics in a useful way which is what was mean when this was written:

Richard Hamming, "Mathematics on a Distant Planet", American Math Monthly, 105:7, 640–650. (Aug - Sep 1998)​

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Just as infinity isn't a number in any number system you have described and certainly isn't an integer, "infinity-th" isn't a word.

 

So, you mean to say that a point has an infinitesimal diameter but no center-point.

Consider any two such distinct points A and B at a distance x in the 3D space.

How you will measure the distance between these two points?

From outer periphery to outer periphery or inner periphery? or from center to center?

 

Do you think 'a point' is a constituent of a line?

 

The intersection of two non-parallel lines is a single point. That point is common to both lines. Therefore points are anonymous constituents of lines that do not spring into existence when lines cross but rather are identified, named in a sense, and lose their anonymity when so identified.

Rather than saying a line is "made up of points" which leads to the misleading mental picture of constructing a line point-by-point, I would say that the fundamental object of all of a particular mathematical space are points that have various spatial relationships with each other and a line is a unique locus of all points that share a property of co-linearity. That is any two distinct points define the co-linearity condition that establishes which points are in the line and from any two distinct points in that particular line the same co-linearity condition is recoverable.

This viewpoint is entirely consistent with point-set topology and analytic geometry in which "existence" has a meaning of "existence within the mathematics framework" which is naturally distinct from any notion of physical existence.

 

And what money did they come into the world with? If you stretch that back enough generations, you see that at some point money did come from nothing. Where was all that money when we lived in caves? Or even before there were humans?

 

If you wish to refer to ex-nihilo in such a oblique fashion then please accept a response that follows in the same manner. There is no "before time" as this is illogical. Likewise there can be no "after time" so it follows that time is eternal both past and future...[from the only perspective available to us, that being existent]
So.... therefore the money always existed. [ and the laws of thermodynamics says so as well ]

 

Well now we're into Adam Smith territory. He wrote The Wealth of Nations to try to understand where it is that wealth comes from. But if you're saying that all the stuff we have right now ... the global economy, the iPhones, the airplanes, the container ships, all the money in the world ... if you're saying it already existed when we lived in caves ... I would just ask you, where was it? Did a caveman say, "We'd have iPhones by now but nobody's invented capitalism yet. Or electricity."

http://en.wikipedia.org/wiki/The_Wealth_of_Nations

 

I might add that "ex-nihilo" or "from nothing" does not necessarily demand that nothing "precedes" something as the universe may very well be a continuum of "instantaneous" emergence from nothing. The need to "move" [energy or time] being that emergence. As movement [time] resolves the paradox of zero dimension in 3 dimensional space IMO.

 

It is an interesting topic but not one for this thread...
However "wealth" is about time [energy] efficiency due to time effectiveness. This does not effect whether a person is born with money or not but whether the money is sufficient or in excess of demand.

 

How will you ensure or prove the continuity of a line?

 

Continuity is assumed in Euclidean geometry (Tarski's axioms for Euclidean geometry explicitly cover continuity), it is proven in analytic geometry from the properties of the real numbers. In point-set topology the line is the locus of all points that share the property of co-linearity and so continuity of the line is a function of continuity of the space.

 

I think these are just continuity of a function rather than continuity of the line consisting of points.

Is the continuity of space just assumed or there is also a proof for it?

 

Then you are arguing from a position of ignorance and thus fail to be convincing.
Axiom 11: Tarski's axiom of continuity:

\(\exists a \,\forall x\, \forall y\,[(\phi(x) \; \wedge \; \psi(y)) \rightarrow Baxy] \rightarrow \exists b\, \forall x\, \forall y\,[(\phi(x) \; \wedge \; \psi(y)) \rightarrow Bxby]\)
"If \(\left{ x : \phi(x) \right}\) and \(\left{ y : \psi(y) \right}\) are subsets of points and a is a point such that every choice of a, x and y are co-linear with x between or identical to a and/or y, then there always exists a point b which for every choice of x and y is co-linear and between or identical to x and/or y."​

What that is saying is that there are no next-to points. because if \(\phi(x) \leftrightarrow x = d\) and \(\psi(y) \leftrightarrow y \; \textrm{colinear with} \; d \; \textrm{and} \; a \; \textrm{but further from} \; a \) then point b has to be between or equivalent to d or some closest member of \(\left{ y : \psi(y) \right}\) . Assuming \(b \neq d\) leads to a contradiction because by the density axiom, there must be at least one distinct point between distinct points b and d. Therefore there is no closest point of \(\left{ y : \psi(y) \right}\) and therefore \(b=d\).
Axiom 22: Tarski's axiom of density:

\(x \neq z \rightarrow \exists y \left[ x \neq y \; \wedge \; y \neq z \; \wedge \; Bxyz \right]\)
"If x and z are distinct points there is always a third distinct point somewhere between x and z."​

http://www.math.ucla.edu/~asl/bsl/0502/0502-002.ps

The continuity of mathematical spaces like \(\mathbb{R}^3\) are built-in to the definitions. The continuity of physical space was assumed starting with at least the ancient Greeks.

 

This idea of the integers being arbitrarily spaced points along a real number line changes when you introduce rational numbers. Then you have to have operations that preserve real distances, since you want half the distance between any two consecutive integers to be the same everywhere, for instance, and 1/nth the distance likewise.

We can count things that aren't unit distances apart, so the natural numbers (ordered by counting or adding "one more") and by extension the integers are just discrete objects, kind of 'sprinkled about' like sheep in a paddock. When rational numbers appear, the integers are necessarily the same 'rational' distance from each other. So the integers get 'regularised' over the rational numbers.

 

What you are saying is that 1+6 = 7 and 8/2 = 4 in both the integers and the rationals, but the rational number 1 is conceptually different from the integer number 1 because 1/2 is only a legal expression in the rationals. I agree.

This conceptual difference is fully captured by the formalism of the construction of working models of the rationals from the natural numbers.

The natural numbers 2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1, etc have a number of representations. The usual one is the finite ordinal numbers. An alternate would be the Peano axioms.
The integers are constructed as a pair of natural numbers z=<a,b> and we apply the rule that \(z_1 = z_2\) if \(a_1 + b_2 = a_2 + b_1\).
The rational numbers are constructed from a pair of an integer and a non-zero natural number: q=<z,n> and we apply the rule that \(q_1 = q_2\) if \(z_1 \times n_2 = z_2 \times n_1\).
The reals may be constructed by either Dedekind cuts of all the rational numbers or Cauchy sequences over the rational numbers.
Seen in this way, construction of the rationals or reals can seem intimidating. But from these formal constructions we can prove \(q_1 \times (q_2 + q_3) = q_1 \times q_2 + q_1 \times q_3\) from first principles and having done this once for the axioms of arithmetic we can be certain our use of the rationals or reals doesn't have logical pitfalls.

And having experienced different concepts of number, you can see that my understanding of the real numbers rests on their definitions. You can't argue with the definitions of a mathematical object because to change the definitions changes the object unless you can prove the change is no true change at all. So all this talk about the smallest positive number makes no sense with the definition of either the reals or the rationals or even more exotic number systems like the hyperreal and surreal numbers.

 

The 'assumption of ancient Greeks' is not a valid proof for 'continuity of points' in the physical space.