Zenos paradox shows the infinity within the finitude of objects. The fact we can break a candy bar in two shows this applies to our world. That is enough for banach tarski. You can take infinity out of infinity. The extra cantor stuff well extra

You said points have no size. I do not see how any part of time could have no size. If it has no size, then no time is passing at that "point", therefore it is not part of time. The same principle holds for space. If it has no size, then it cannot be part of spatial existence, because there is no space there. It is very clear to me, that if points have no size, then they are excluded from space and time, because things existing in space and time have size. Having size is what makes them spatial-temporal. Do you not understand this? — Metaphysician Undercover

I do not understand why you think that things that exist in space and/or time must have size. Why is it impossible for something to exist (in space and/or time) and not have size?

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Some perfectly sensible and familiar rational numbers, such as 1/3 = .3333333..., have infinitely-long decimal representations. — fishfry

This depends on the meaning of the symbol "0.333~". According to the way most people define "0.333~", it is not true that "1/3 = 0.333~". By standard definition, 0.333~ is a number smaller than 1/3 (in the same way that 0.999~ is a number smaller than 1.) It's not a decimal representation of it. "0.333~" does not mean "the limit of 0.333~".

Now people are saying objects have no size. Oh boy!

Bertrand Russell on the Axiom Of Choice: "At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it."

I can't even hazard a guess as to how you think "most people" define "0.333~" (I am more accustomed to the ... notation, but I assume you mean the same thing).

Just as I described, if it takes up no time, it is not "in time". Things which exist in time, have temporal extension, that's what existing in time means. Likewise things which exist "in space" have spatial extension, that's what existing in space means.

If you're having difficulty understanding what this means, then try to imagine a point in time which has no temporal length. At this point, no time is passing. When no time is passing, this is not "in time" because time is always passing, that's what time is; and if time were not passing, we would no longer be "in time". Likewise, think about a thing which occupies no space, All the space around us is occupied by objects, air, etc. A thing which occupies no space could be anywhere, and everywhere, or even nowhere, all at the same time. Bit "existing in space" means that the thing has a particular spatial location, so a thing with no spatial extension cannot be "in space"..

Of course definitions can be contradictory. — tim wood

Contradictory definitions are rejected on the basis of the law of non-contradiction. So it is pointless, and rather ridiculous to put forward a definition which is contradictory, and expect that someone ought to accept that definition.

↪Magnus Anderson I can't even hazard a guess as to how you think "most people" define "0.333~" (I am more accustomed to the ... notation, but I assume you mean the same thing). — SophistiCat

I avoid the "..." notation because it looks ugly when used in forums without LaTeX support. But yes, that's what I mean.

"0.333~" represents the infinite sum 3 x 1 / 10^1 + 3 x 1 / 10^2 + 3 x 1 / 10^3 + ... + 3 x 1 / 10^inf. It does not represent its limit.

The Beatle song Here Comes the Sun has been analyzed over and over by music experts, but anyone can listen to it and get stuff out of it from their own minds. Likewise anyone can see that objects in the world are both finite and infinite. And it must be, because we are made of matter and there must be a contradiction in it. https://en.wikipedia.org/wiki/I_Am_a_Strange_Loop

Godel says that mathematics is either contradictory or incomplete. I think it's clear that there is a contradiction in the first step of geometry

, MathJax is supported:

$\displaystyle\frac{1}{3} = \displaystyle\sum_{n \in \mathbb{N}} \frac{3}{10^n} = 0.333\cdots$

where $\mathbb{N}$ does not include $0$

To render:

Anyway, to 's point, the three expressions around the $=$ symbols are just different ways of writing the same number, out of any number of ways.

If I were to to move to the bed, I have to move half that. If there is no half, then I am already there! There is NOTHING that stops this process. Calculus merely distracts people from seeing the contradiction at the very foot of geometry, the contradiction that makes geometry possible. Zeno took his reasoning further than perhaps even Parmenides would have, and confounded the Pythagoreans even further than irrational numbers did.

Now people are saying objects have no size. Oh boy! — Gregory

Whether they can be said to exist or not, these are abstract objects, not my sandals. :)
The formalisms, theorems, etc, is how you treat them, you don't wear them on your feet.

Say, there's 10 meters over to the neighbor's front door.
That's a distance between two places here in the world.
Maybe some prefer saying "there's roughly 10 or 11 meters over there"; doesn't really matter much.
Unless you walk the wrong way, then it's almost 40,000 km longer.
The mathematical treatment (or modeling) of these things hold up just fine.

There are only two classes of people who need to carefully make this distinction: mathematicians, who are trained on this topic in their undergrad years; and philosophers, — fishfry

Mathematicians? Not necessarily. "Flaws" . . . not necessarily. Incidentally, your compact form of Leibnitz expansion has a simple error. And 1/3 =.333... = limit of a geometric series, well defined. You may be talking about mathematicians who labour in foundations. Making such fine distinctions is unnecessary in most math careers, IMHO.

Zenos paradox shows the infinity within the finitude of objects. The fact we can break a candy bar in two shows this applies to our world. That is enough for banach tarski. You can take infinity out of infinity. The extra cantor stuff well extra — Gregory

Once again, I can't argue with this. :brow:

"0.333~" represents the infinite sum 3 x 1 / 10^1 + 3 x 1 / 10^2 + 3 x 1 / 10^3 + ... + 3 x 1 / 10^inf. It does not represent its limit. — Magnus Anderson

What do you think an "infinite sum" is then if not the limit (if it exists) of the partial sums?

The standard view of the positional notation is that it is a representation of a number as a series, with digits serving as coefficients in front of the base, and their position designating the power of the base (positive before the dot, negative after the dot). But I still have no idea what you think "most people" think of it.

As you say, an infinite sum may not have a limit. If you say that the concept of infinite sum and the concept of limit are one and the same concept then how is it possible for an infinite sum to have no limit?

Furthermore, the difference between the two concepts is bigger than that. For example, two infinite sums that approach the same value can represent different quantities. For example, $0.9 + 0.99 + 0.999 + \dots$ represents a number greater than $0.5 + 0.25 + 0.125 + \dots$ even though they both aproach but never reach $1$. Indeed, there are numbers greater than $0.999\dots$ but lower than $1$. Hexadecimal $0.FFF\dots$, for example, lies somewhere between the two numbers.

Thank you very much.

Oh boy, you are one of those 0.999... =/= 1 people. Never mind then.

Likewise anyone can see that objects in the world are both finite and infinite. — Gregory

I've never seen an infinite object. In fact, I really don't see how an object could be infinite. What we sense and apprehend are the boundaries of an object. Without such boundaries what we'd be perceiving could not be apprehended as an object. And since we perceive boundaries it's questionable that we could even apprehend the infinite. If "infinite" is to even make sense as a concept, it cannot refer to an object

The infinite divisibility of objects, which even Aristotle admitted was real, means all objects are infinite within finite bounds. My claim is that this is a basic contradiction at the base of material reality and geometry. As in "God does not play dice", God does not create the world as a mathematician, but as a poet

Metaphysician Undercover;

You should be aware that the mind is image oriented. It creates, analyzes, and stores images. Vision is the dominate sensory input from the external world. It's the nature of the mind to form the simplest images possible to represent things outside the mind.

The purpose of abstraction is to eliminate detail irrelevant for our purpose. A simple example, children use this form of abstract representation with their 'stick' figures for people. Thus we use ideal lines, circles, cubes, etc. to convey information to others. A social benefit is realized via storytelling.
This is most obvious in numbers used for counting, assessing the multiplicity of a collection of things. The numbers exclude all attributes of the things being counted.

As for the 'point', it too is an abstraction to serve as a location/coordinate. A surveyor places a stake as a marker/point for a property line. Being dimensionless, you can't see it, but an object is provided in the form of a marker, blob of medium on a surface, pixel on a screen. The point is somewhere within that marker/blob/pixel. This is not a problem since any calculations requiring the location will not vary to any significant degree. The same situation for the 'line' having no width. In graphics it's a continuous marker, in surveying it might be a laser. The line formed from points is a contradiction of terms since a point has no extent. How many zeros are added to a register to accumulate 1? We don't see trajectories or orbits either, but they still serve a purpose.

The continuum is another story.

An interesting quote by Poincare, The Measure of Time, 1898

"We helped ourselves with certain rules, which we usually use without giving us account over it [...] We choose these rules therefore, not because they are true, but because they are the most convenient,...
In other words, all these rules, all these definitions are only the fruit of an unconscious opportunism.“

I think we can know what an orange is, and that it doesnt have less volume when cut in half. It can be so divided infintely, so it's infinite AND finite. This is so obvious

If you infinitely divide an object and line the pieces up largest to smallest, what is at the far end?

surveyor places a stake as a marker/point for a property line. Being dimensionless, you can't see it, but an object is provided in the form of a marker, blob of medium on a surface, pixel on a screen. — sandman

If the survey stake is the marker point, then the point is not dimensionless, and clearly can be seen. But if the survey stake simply represents a point, and the point represented is dimensionless, then the dimensionless point is not a location at all, because the survey stake marks the location.

What you say here, that the survey stake is the location point, and also that it is dimensionless and can't be seen, doesn't make sense, because clearly the survey stake can be seen.

I think we can know what an orange is, and that it doesnt have less volume when cut in half. It can be so divided infintely, so it's infinite AND finite. This is so obvious — Gregory

This is nonsense. It's actually very obvious that an orange cannot be divided infinitely.

I think the notion of an orange being divided infinitely is ap-peeling. :smirk:

I'm so happy that Cauchy, Weierstrass, and others settled this issue for mathematical analysis long ago. It made my career so much easier. :cool:

But there is no way to mark an exact irrational length on the ruler - unless a line representing an irrational distance is constructed (like the square root of two) and marked on the ruler by direct measurement. Correct? — tim wood

Depends on what you mean by mark. Every point on the real line "marks" that point. All the irrationals and even the noncomputable reals. Every real number is the location of some point on the real number line.

Of course if you mean construct as in compass and straightedge, that's different of course. But every point is at some exact real number location to the left or right of the origin. All the points are marked, the way I see it. Perhaps we're talking semantics.

Continuum is a set of points where for every two points in the set there exists a point in the set that is in between the two points. — Magnus Anderson

I happened on this remark which you made a while ago. I wanted to note a correction.

Consider the rational numbers. They have what's called a dense linear ordering, which means that between any two rationals there exists another rational. Just take their midpoint, for example, which must be rational.

But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line.

That is not a continuum, mathematically or morally.

What characterizes a continuum is the least upper bound property. This says that every nonempty set of reals that is bounded above, has a least upper bound.

The least upper bound property is false for the rationals. For example the set

$S = \{x \in \mathbb Q : x^2 \lt 2\}$

where $\mathbb Q$ is the set of rationals, does not have a least upper bound in the rationals. But it DOES have a least upper bound in the reals.

That's exactly why the reals are regarded as a mathematical continuum, and the rationals aren't.

Note please that I'm only saying what a mathematical continuum is. I'm not addressing any of the many philosophical objections there could be to calling the real numbers a continuum, But mathematically, the reals are a continuum and the rationals are not.

https://en.wikipedia.org/wiki/Least-upper-bound_property

Incidentally, your compact form of Leibnitz expansion has a simple error. And 1/3 =.333... = limit of a geometric series, well defined. You may be talking about mathematicians who labour in foundations. Making such fine distinctions is unnecessary in most math careers, IMHO. — jgill

Oops thanks I left out the exponent of the -1. Fixed it.

Not sure I followed the rest of your remark. My post is intended to clarify the thinking of all who say that "irrational numbers introduce infinity into mathematics." This is actually false. It's noncomputable numbers that introduce infinity into mathematics. A far more subtle and philosophically interesting point.

In any event I'm not arguing that working mathematicians care about these fine points in their daily work. I'm simply noting that a number is not any one of its representations; and that having an infinite decimal expression is an artifact of radix notation and not the real numbers themselves. After all pi and the square root of 2 have perfectly finite Turing machines that generate their decimal representations. That's the key point. Not who thinks about what during the work day.

I don't see why Zeno's paradox is not a paradox but Banach-Tarski is. — Gregory

Naming conventions are a matter of historical accident. Banach-Tarski is a theorem. It's not actually a paradox. It is however a veridical paradox, which is a true result that is contrary to intuition. If you want to call it the Banach-Tarski theorem that would be both accurate and less confusing. It's not really a paradox. It does show the distinction between abstract math and physics. The kinds of set-theoretic manipulations required in Banach-Tarski can't be done in the real world ... as far as we know, anyway.