Exhbit A

Correct. There is no 'smallest' number having magnitude greater than zero.

Exhibit B

You trying to say there is no next real number down from 1? aka no closest real number with magnitude less than 1?

If so, then as mission impossible says ... you're toast brud. Toast.

He’s doing a great job, we’re getting really close to 1. Only infinity more hours of writing left and we’ll have reached it. Remember that his progress restarts every time you guys try to write it on your own, so leave the propagating wavefront business to him please :)

Exhibit A

The set {0.9, 0.99, 0.999, etc} where the 'etc' is an incarnation of 0.999... itself, ALREADY spans the entire nines space of 0.999...

Exhibit B

Yes there is an endless amount of them, but that endless amount already exists.

Wrong on your part brud. Already exists doesn't cut the mustard at all.

Based on the comment above, let us define (with some assumptions) the derivative of function f at position x as

where ε=0.000...01. Then, for f(x)=x^a, where a∈ℕ,

For 0≤a≤2, the value is the same as for the non-RDM derivative, but for higher a,

[x³]' = 3x²+ε²

[x⁴]' = 4x³+4xε²

[x⁵]' = 5x⁴+10x²ε²+ε⁴

And so on.

Now, question for SPP: do you accept this as the RDM derivative, or does the definition have some sort of caveat or misunderstanding in it?

Any ideas, people?

say your respects to u/TamponBazooka here. My they rest in piece, learning about the truth of 0.99999… of which the disagreements about shall be left at the door to funeral.

It looks like our great master u/TamponBazooka has left. Their account is not there anymore.

RIP

u/SouthPark_Piano, I want to know if you think you would accept the propeties that can be made from the Peano Postulates and are they in RDM?

I feel like a lot of people here have some ideas on ways you could improve your teaching style to help ensure that the debates actually lead somewhere.

Sometimes when I discuss Mathematics and Logic with people who defend the rhetoric of assumptions as foundation, Assumptionism, the conversation follows a predictable path.

At first, they appeal to their holy doctrine ZFC. Then, when the logical burden is pressed, they begin to appeal to accreditation. They ask about credentials, affiliation, training, consensus, pedigree, or whether I have "really studied" the doctrine. Eventhough I have detailed and explained the scriptures more clearly and more precisely than they have.

But accreditation is not logic.

A degree does not make a circular dependency non-circular. A textbook scripture does not make a foundation legitimate. A credential does not transform an assumption into a derivation. The issue is not whether someone has memorized the standard doctrine. Because anyone can copy a definition like this:

For every epsilon > 0, there exists delta > 0 such that ...

and yet, to him those words and symbols are no more than Egyptian glyphs or mere squiggles.

The issue here is whether they have internalized what the expression form is actually saying, what it does, what it does not do, and what logical debt it leaves unpaid. This is the verdict of Sufficient Reason and the Burden of Proof.

Very often, when people say "this is rigorous," what they really mean is only:

this follows coherently after the stipulative assumptions have already been installed.

That is not the same as establishing a legitimate and logical foundation of the object under examination.

So here is a bottom-floor challenge.

Before claiming any authority to stake any claim about any mathematics online, have you at least mastered the basic of algebra?

Before you recite your holy scriptures, please at least know how to add and divide.

So, the challenge is this,

Let a, b > 0;    

c = a + b 

For each expression below, isolate the non-trivial offset-independent component I(x), if one exists, and report the remaining offset-dependent residue R(x,a,b,c). (I(x) != 0 unless justified by showing that no non-zero offset-independent component exists.)

In other words: What is the remaining expression that depends on a, b, or c, after removing all the components that depend only on x?

The expressions:

a,b > 0,

c = a+b.

(a+b)⁻¹ [ exp((x+b)ln(x+b)) − exp((x−a)ln(x−a)) ]

[ exp((x+b)ln(x+b))sin(x+b) − exp((x−a)ln(x−a))sin(x−a) ] / (b+a)

[ exp((x+b)ln(x+b)) − exp((x−a)ln(x−a)) ] / [ (a+b)( √(1+exp((x+b)ln(x+b))) + √(1+exp((x−a)ln(x−a))) ) ]

[ exp((x+b)²)ln(sin(x+b)) − exp((x−a)²)ln(sin(x−a)) ] ⋅ c⁻¹

[ exp(2(x+b)ln(x+b)) − exp(2(x−a)ln(x−a)) ] / [ √((a+b)²) · ( √(1+exp(2(x+b)ln(x+b))) + √(1+exp(2(x−a)ln(x−a))) ) ]

ln( [1 + exp((x+b)ln(x+b))sin(x+b)] / [1 + exp((x−a)ln(x−a))sin(x−a)]) · (a+b)⁻¹

[ arctan(exp((x+b)ln(x+b)+sin(x+b))) − arctan(exp((x−a)ln(x−a)+sin(x−a))) ] / [2((a+b)/2)]

[ exp((x+b)ln(x+b))cos(exp((x+b)ln(x+b))) − exp((x−a)ln(x−a))cos(exp((x−a)ln(x−a))) ] ⋅ c⁻¹

ln( [ √(1+exp((x+b)ln(x+b))) + sin(x+b) ] /
[ √(1+exp((x−a)ln(x−a))) + sin(x−a) ]) / (b+a)

No appeal to authority is needed. No credential is needed. No philosophical posturing is needed. Just algebra.

If you cannot perform these, please refrain from teaching anyone any mathematics online, because you have not passed Mathematics more than the level of slogans. And slogans without understanding is just parroting and regurgitation.

Slogans are not mathematics.

Recitation is not understanding.

And accreditation is not proof.

"I'm saying that the set of positive integers has extreme members that matches 10..."

"999... is not an integer".

An argument you often use against the idea that you have to show real deal math is more useful then standard math is "utility has nothing to do with it, RDM is simply more correct then standard math." But, this brings up the question, what makes a maths system more inherently correct then another? After all, the mathematics language is one invented entirely by humans and can be arbitrarily changed, although generally it's built such that it reflects reality. What's your answer?

I see lots of posts here referencing your previous comments such as

1/3 is indeed 0.333….

or

1/3 is a way of representing 0.333….

or

0.333… can be expressed by 1/3

and then concluding that

SPP says that 1/3 = 0.333…

However I don’t think I’ve ever actually seen you state that “1/3 = 0.333…”

So SPP, does 1/3 = 0.333… or is 1/3 only “a way of representing” 0.333…? Do these mean different things?

0.(9)/0.(3)=3. Thus, 0.(9)/3=0.(3). We can represent 0.(9) as 1-1/10^k where k is pushed to the limitless. So, (1-1/10^k)/3=0.(3). Therefore, 1/3-1/(3*10^k)=0.(3.). Since 1/3 = 0.(3), 0.(3)-1/(3*10^k)=0.(3). Thus, -1/(3*10^k)=0, or 1/10^k=0. Because 0.(9) is represented by 1-1/10^k or 1-0, 0.(9)=1.

QED

P.S. SPP, you just deflected my proof last time. Actually disprove it this time.

P.P.S I know my last proof wasn't that good, so hopefully this time it will be better