Without getting too involved in the semantics of hyperreals, they do allow us to demonstrate a difference between 0.99999... and 1. The distinction is between limit and st(limit) if I read it correctlywtf wrote:I skimmed the Better Explained piece. It seems like a compendium of all the wrong and misleading things people say on these .999... discussion forum threads.

In particular, the author claims that .999... < 1 in the hyperreals. This is simply not only false, butprovably impossible. Professor Amit explained whyin the very Quora article that you yourself linked.

Let me explain how this works, although this goes beyond what Prof. Amit said.

Consider the first-order theory of the real numbers. This is the formal syntactic list of axioms involving addition, multiplication, order, etc. It does not included the completeness axiom, which is second order (we quantify over subsets and not just over elements).

The standard reals are a model of this theory. So are the hyperreals. It's a theorem of model theory (the theory of formal systems and their models) that all models must satisfy the same first-order theorems.

.999... = 1 is a first-order theorem of the theory of the real numbers. That means it's provably true (ie there is a logical proof from the axioms) in the realsand also in the hyperreals.

It can not possibly be otherwise because both the reals and the hyperreals are models of the same set of axioms. So any proof that works in the standard reals also works in the hyperreals.

Now I am NOT saying that infinitesimals aren't a useful intuition; or that ultimately the physicists may say some day 200 years from now that we can detect infinitesimals in a particle accelerator.

What I am saying is that .999... = 1 is a theorem in the hyperreals; and that this MUST be so.

Finally, what about second-order properties such as completeness? Well, the hyperreasls are not complete. They are shot full of holes. In my opinion that's a very poor model for the intuitive idea of a continuum. The hyperreals are a tremendous distraction in the .999... discussion but a very common one.

the limit of the sequence 0.9, 0.99, 0.999, ... is less than 1 in the hyperreals whereas the st(limit of the sequence 0.9, 0.99, 0.999, ... ) = 0. The st operation acts as a rounding up to the nearest real number operation similar to the ceiling function in conventional math.

The real analysis book I have says this about the decimal representation:

1) decimal representation represents an infinite series

2) an infinite series represents an infinite sequence

3) the least upper bound of the sequence represents the limit of the infinite sequence

4) the limit of the sequence represents the value or sum of the sequence

5) the "=" sign represents the value

Therefore 0.9999... = 1 according to that

http://imagizer.imageshack.us/a/img921/1559/gQBd4b.jpg

"If lim S exists, we say that this series is convergent and

**call this limit**the sum or the value of this series." That's about as deep as I saw them go with it, I'm assuming this is the axiom.

Pluto used to be defined as a planet, now it's not. If I were to have asked "Is Pluto really a planet?" back then I would not be inquiring about what it is currently defined as or what I was taught in school and pounded into my head all my life. I would be asking whether or not it is justified that Pluto is classified as a planet. The real question is whether or not 0.9999... should be classified as the limit of a sequence.

And found this definition of "<", ">" and "=" in the book

http://imagizer.imageshack.us/a/img924/8806/Ksl4AO.jpg

They use the subtraction logic and if the difference is positive then the "a>b" or "a<b" is used according to that. But they did say that when defining "a>b" that a and b must both be elements of the real numbers. This would require a proof that 0.9999... is a real number and then determining whether the axioms of that proof are justified or not as well.

The argument for that is 0.9999... is 1 because there are no real numbers between 0.9999... and 1 therefore 0.9999... and 1 are the same because they do not differ by a real number. All real numbers differ by other real numbers where it's assumed that 0.9999... is a real number. But if 0.9999... is not a real number because 1 and 0.9999... do not differ by a real number but do differ by something then 0.9999... is not a real number could make sense too to explain why no real number exists between them rather than just assuming they're equal. Claiming 0.9999... = 1 because a real number cannot be produced between them commits an argument from ignorance fallacy https://www.logicallyfallacious.com/too ... -Ignorance

Argument from Ignorance

Ad Ignorantium

(also known as: appeal to ignorance, absence of evidence, argument from personal astonishment, argument from Incredulity)

Description: The assumption of a conclusion or fact based primarily on lack of evidence to the contrary. Usually best described by, “absence of evidence is not evidence of absence.”

Logical Form:

X is true because you cannot prove that X is false.

X is false because you cannot prove that X is true.