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Comment by yequalsx | Hacker News Reader

Property of real numbers: between distinct real numbers is at least one other number. Now try to find a decimal representation of a number bigger than 0.9999999... but less than 1.0. You clearly can't. They must be equal. No need for infinitesimals.

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Okay, so it follows from completeness. Now prove that real numbers are complete (or provide a construction of the reals that uses completeness as an axiom) without using concepts foreign or confusing to someone with a middle school level exposure to math.

It depends on where you want to start your axioms. We can go the Whitehead/Russell route or just use this as an axiom.

We convince children that 1+1 =2 without delving into the Peano axioms. It's ok to not delve too deeply into the axiomatic structure of the reals.

You're just proving their point. None of the things you talk about are even remotely obvious or “natural” to people we're talking about. You want them to ”choose axioms”? Axiom-a-whaaa? 1+1=2 does not need Peano axioms because it's cognitively fundamentally different than 0.999...=1.0. It is immediately obvious because dealing with simple arithmetic on natural numbers is in everybody's experience constantly. In other words we do not need to convince them. Clever untrained people would be able to get 0.999... thing quickly if you gave them some explanation and let them think a bit, if they cared, but properties of real numbers are far from obvious or basic.

> Okay, so it follows from completeness.

There is nothing to do with completeness here; 0.999… = 1 is a statement about a series of rational numbers summing to a rational number, and the convergence of the series is established by the fact that it sums to the right-hand side, not by an abstract appeal to completeness.

I'd argue that the distance is 0.00000... = 0 and that distance = 0 implies that the points are equal.

Doesn't that property come from the fact that there is no infinitesimal in real number?

> find a decimal representation of a number

0.99999999...1

This denotes the abstract idea that we take 0.999... (infinite number of 9's) and add another digit.

This is no more or less abstract than 0.999... to begin with.

0.999... is a unicorn, and 0.999...9 is a unicorn with a pink ribbon on the tip of its horn.

> add another digit

But you can't. All of the places where you might want to add a digit are already occupied by 9's.

So you're saying a countable infinity can be occupied, so that there is no room for one more.

You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.

> So you're saying a countable infinity can be occupied, so that there is no room for one more.

No. I'm saying that a countable infinity doesn't have room for one more at the end. There is no end.

You can obviously stick a 1 in the middle, but then you have a number strictly less than 0.999...

You might be able to do something funny with a series whose terms are indexed by ordinals. The last `1` would then be indexed by `omega` (the first transfinite ordinal). I can't claim that the ordinary rules of arithmetic would make much sense, but it's certainly an idea that you can attempt to formalize.

It has been done:

https://en.wikipedia.org/wiki/Infinitesimal

Of course you can. Consider the sequence of rational numbers in (0,1). Now consider the sequence of rational numbers in (0,1]. The latter has had an extra term appended.

(This is of course flawed, but I think it illustrates that the question isn't completely trivial. It requires us to carefully distinguish between the notion of an ordered set and a sequence and even then we'll have to deal with the fact that the rationals can be made into a sequence, but not with the same ordering.)

No. There is no such thing as the sequence of rational numbers in an interval. The rationals have a total order, and they can be enumerated, but the total order of rationals does not define an enumeration of the rationals and hence the rationals are not a sequence. If you want to dispute this, tell me: what is the next rational after 1/2?

I took some liberties with the word "the", but I'm pretty sure that you know that it's easy to construct an ordered sequence of all rational numbers using any of a number of standard diagonalization arguments. I'll skip the details you already know.

Eventually, yes, you're going to be able to poke a hole in my argument. It is definitely flawed. I don't know how long it'd take us to get there, but it doesn't really matter. But we're already at the point where this cannot be considered "basic", and that is my true point here.

Attempting to demonstrate that 0.999... = 1 while meticulously avoiding any rigorous definition of what 0.999... means is not very easy and will require you to fend off all sorts of potential jabs from various directions. It's much easier to just talk about infinite sums and be done with it.

> It's much easier to just talk about infinite sums and be done with it.

Indeed. But it's a lot less fun :-)

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