Math controversy should not exist

*If you aren't sure why this implies that infinitesimals can't exist, if a positive infinitesimal is defined as a number greater than zero but less than 1/n for every natural number n, (which 1 - 0.99999... would be if it were positive) than if x is a positive infinitesimal, x < 1/n for every natural number n, and so 1/x > n for every natural number n, a violation of the Archimedean property.

Lemma The set N of positive integers N = {0, 1, 2, . . .} is not bounded from above.Proof Reasoning by contradiction, assume N is bounded from above. Since N ⊂ R and Rhas the least upper bound property, then N has a least upper bound α ∈ R. Thus n ≤ αfor all n ∈ N and is the smallest such real number.Consequently α − 1 is not an upper bound for N (if it were, since α − 1 < α, then α wouldnot be the least upper bound). Therefore there is some integer k with α − 1 < k. But thenα < k + 1. This contradicts that α is an upper bound for N

archimedean.pdf (upenn.edu)

And joke or not. .....It's actually a reasonable consideration, relative to reductionism.

Archimedean Property does not consider laws of thermodynamics. Archimedean property applies a rule of scale and proportion which is not by fact accurate by laws of thermodynamics or in line with human margins of error. 0.9 is proportional to 0.999999 they are equal though are not secureas equal by precise measurements and are subject to change, such low values are demonstrated to change by the choice of letters assigned in Calculus near theend of the English Alphabet.

Not everyone has or can work an Electron Microscope to establish that there is no natural value of 0.999999 and those values created by computers and calculators are fabricated and not calculated by poor practice in calculation. There are several examples of this throughout the history of mathematics as science has evolved over the years. The formula to write 0.9999 is more complex thus taking more work than that what is for most people is not needed to do to acquire the sum which will suit a given purpose during normal math routines...

Not only does not everyone have an electron microscope, no one needs one to verify that the proof that you quoted (which is the standard proof of the Archimedean property) is valid. Also, errors in computers and calculators are irrelevant. No computer or calculator is required in this proof.

And I easily demonstrated that 0.9... will always fall .1 or .01 etcetera, short of 1.

"Everyone needs to have an Electron Microscope to observe as a witness the error as it is made in mathematics and how it occurs ,or it is otherwise seen as human failure only."  

The idea is not to disprove Archimedean Property, the objectis to establish it is not mathematically relevant. Archimedean Property are classifications of subsets and mathematic elements. The Archimedean Property describes that ( 9.0  ≠ 10 ) as ( 0.9 ≠ 1.0) stating that as fact9.0 and 0.9 are normal subgroup( ⪧) the idea is not to disprove Archimedean Property the object is to establish it is not mathematically used relevant. Archimedean Property are clasifications of subsets and mathematic elements. The Archimedean Property describes that 9 is boundary.

In abstract algebra, a normal subgroup (also known as a invaariant suibgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words,a subgroup N of the group G is normal in  Gif and only if gng ^ -1 element (∈) N for all g  element (∈ )G and n element ( ∈ )theusual notation for this relation is N  normal subgroup (⪦ )G

And I easily demonstrated that 0.9... will always fall .1 or .01 etcetera, short of 1.

No rigour required.

The fact that the set of all integers contains more terms than the set of all even or odd integers renders the word more (or greater than) meaningless.

The Archimedean property states that if x is a real number there exists some natural number n such that n > x, so the Archimedean property has nothing to do with subsets or normal subgroups, nor does it "describe that 9 is a boundary." Also, yes, 0.9 is not equal to 1, nor is 9 equal to 10, but 9 and 0.9 aren't "normal subgroups" of anything. A normal subgroup is a type of subgroup, which is in turn a subset of a group that is also a group. 0.9 and 9 aren't groups.

All subgroups of the real numbers are normal, so I don't see this as relevant. This is in fact true of any Abelian group. (A group that satisfies the communatative property.)

These sets are all, rather counterintuitively, the same size. There are, however, sets that are bigger. This video explains it better than I ever could.

And I easily demonstrated that 0.9... will always fall .1 or .01 etcetera, short of 1.

No rigour required.

0.9 = nine tenths.

Therefore ten tenths = 1.

0.99 = 99  hundredths

Therefore a hundred hundredths = 1

0.999 = 999 thousandths.

Therefore a thousand thousandths = 1

And so on.

The error here is the choice in numerical decimal systemthat creates the 0.9999... in the first place. This is not the only error in mathematics,or the only location is a string of formula where error occurs. In post #14 we established only that 0.9, 0.99, 0.999, 0.9999, etc. are groups of subsets of #1 not that they had been precise. We are also describing the use of duodecimal system to negate fractions decimals like (0.9, 0.6, and 0.3). 

I respectfully disagree on what is stated as the Archimedean Property uses the symbol ∈ describing a membership betweenreal numbers the letter ( a ) can describing in calculus a value which is fixed and the {  } brackets identify a set.

meaning group,

Some subgroups are Irrational numbers or Natural numbers they are not all "normal." We can compare Time, Degree, Prime, Natural, Irrational, Rational, Odd, Even, etc. and see this.

In abstract algebra, a normal subgroup (also known as a invaariant suibgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words,a subgroup N of the group G is normal in  Gif and only if gng ^ -1 element (∈) N for all g  element (∈ )G and n element ( ∈ )theusual notation for this relation is N  normal subgroup (⪦ )G

Since infinite steps are needed, infinite time is needed as well.

There is no such thing as a group of people as sets of infinite people infinite describe all the people as one normal group.

The bus as explained in the video is stating that a lie is taking place somewhere.

If bus (a), being a set value in calculus is in fact infinite there is no bus (b) as set value. Bus (a) is holding all people. otherwise, bus (a) is properly called bus (x) and is an approximation value only.

Now I understand the issue. "Group" and "set" do not mean the same thing in math: group, set. Please familiarize yourself with these definitions.

We must be misunderstanding each other here. I agree with this.

they will probably not say "3.141592...." Instead, they will probably say "π.

If S is any subset (not just a subgroup) of the real numbers, then gng^-1 = gg^-1n = n ∈ S by definition of g^-1 and by the commutative property for any n ∈ S and g ∈ G, G here being either the multiplicative group of real numbers, or the additive group of real numbers, whichever you were referring to.

so I'm not sure where you are getting this from

This sentence isn't the most readable, but if I'm interpreting it correctly, you are saying that an infinite group of people. That is true. The video is just using something concrete (people) to assist in the explanation. It is not being suggested that any of this could actually happen. Only in the abstract. As the video says, this line of inquiry lead to the invention of the modern cell phone and computer, so if it's all wrong, then I'm not sure how it's possible for us to have this discussion.

Could you please explain what is meant by bus (a), bus (b) and bus (x)?

"I had written."

"I respectfully disagree on what is stated as the Archimedean Property uses the symbol ∈ describing a membership between real numbers the letter ( a ) can describing in calculus a value which is fixed and the {  } brackets identify a set."

1. We have a membership to group by the use of mathematic"∈" symbol means membership and well it is ArchimedeanProperties we are speak of here.

2. The bracket {  }  identifythe  math principle of set in all math notation not just Archimedean property.

3. Archimedes’ work is on calculus which is in general explanationbrings mathematics of calculus in to describe x as a  "little bitof" or most often described as " an element of." 

The whole ofcalculus describes the topic of element as of being more than on so is sets andgroups of elements. 

I'm not going to focus on "Pi" as it was part of a list of mistakes to corect.  I have always felt this error in desperate needing of a solution. The topic at this point may drive us toa lesser understanding at the moment. On a personal note, I have resolved anyissues with a approximation of Pi, writing a new formulation in trigonometry as theterm cPi using work from Isaac Newton's trigonometry, whereas triangles bothright and left 90 degree angles can be used to calculate circumference as hypotenuseof those respected 90 degree angles. as the letter (c) suggest in calculus Ihave a created a formula that sets a constant that is not irrational as value.

In the above exsample did all of the mathematic symbols translate across to this forum.They didn't for me and the math symbols of a triangle with the apex facing right or left  is the principle missing for subgroup here.

If a infinite number of people on / in one objectis to true. All other buses are empty as fact. There are an infinite number of buses but only one bus of infinite people, as there are no people left to go on any more buses but one. "The whole idea on the video is a lie. Not even a good lie at that." Understand?

It is part of a minor math discrepancy about the video. When describing the bus and people in calculus the letters a and b would be written as buses  y and z they are variables not fixed values.

Exactly. { } brackets identify a set, not a group. As I explained, in math, those are two different things. It's an important distinction.

Everything in the video has a fixed value. The cardinality of the hotel and of the buses is aleph null, and the cardinality of the party bus at the end of the video is the continuum. (see the second meaning of "continuum" on the linked webpage) Cardinal numbers never show up in calculus, but this isn't calculus, it's set theory.

I'm not sure I follow. Are you claiming that the decimal expansion of π is invalid? 

Nope.

It just doesn't work.

Because you can not divide 1 precisely by 3 decimally.

The equation fails at the first stage.

.3 is 3 tenths

Times 3 = 9 tenths.

9 tenths does not equal 1

Doesn't matter how many 9's.

 .999 = 999 thousandths

1 thousandth short of one.

The problem: There is no step at which you complete the path. No matter how many steps you make, you will never complete the path. The duration of a step is irrelevant, as the step never lasts 0 seconds. Since infinite steps are needed, infinite time is needed as well.

That, my friend, is the math problem.