I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste.

> does it matter if the proof of a fact is incorrect if the fact itself is correct?

Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a question of correct vs incorrect, I think you're asking for more precision, and/or for a form that meets your own higher standard.

It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.

BTW, reading the blog post you linked to on surreals, the "proof" looks to me to be more hand-wavy than Euler's proof that .9bar = 1. The proof begins by stating there are a finite number of 9s, in direct contradiction to the hypothesis. 10^-inf = 0, so from this blog post I don't yet see any reason why surreals clarify anything here, it feels like the opposite, it feels like obfuscation.

This could be an argument over representation and not the values of numbers. If you start by defining 0.9bar to be a different number than 1 for the specific reason that it's written down a different way, then fine. That's what the surreal "proof" tells me. Euler's proof is talking about the value of 0.9bar in the limit, not the representation. (Even if that's stated without rigorous definitions of limits.) The proof is saying the values of .9bar and 1 are the same. If the surreal number .9bar were strictly less than 1, that must mean there's another surreal number closer to 1, but there isn't, so I don't accept the surreal argument as valid logic, other than playing a semantic trick by saying 'look I defined they way we write a number to be meaningful, therefore 0.999... is by definition different than 1.'

By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."

I'm only going to respond to a few chosen points here because I find your post kind of meandering and hard to follow. I'm not cherry-picking (I don't care about "winning" any argument), I'm just trying to focus my response a bit so it increases the likelihood that you'll understand what I'm trying to say.

> I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste.

> [...]

> It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.

I'm going to try to be more formal, but not entirely formal since (1) the details are almost never-ending and require a lot of formal logic and (2) I'm not 100% sure about the reasoning myself.

What I mean by a "proof" is a sequence of logical steps that start with axioms of your logical system. We are obviously not looking at things that formally, but I actually think it is still an important point. The rational numbers can be thought of as being defined by certain axioms of arithmetic. E.g. you have the natural numbers as well as the minimal extra values so that you can add, subtract, divide, multiply, etc.--in other words you have a field. Let's call these the arithmetic axioms. Then when you go to the real numbers you basically extend the rational numbers in such a way so that you have completeness and retain all the previous properties. So basically for real numbers you have the prior arithmetic axioms and you have the completeness axiom.

Next comes the question of what is actually to be proved. For that we _must_ make some sort of definition of what we mean by "0.9...". Let us define it as the limit of the sequence of partial sums (all of which are rational numbers) _if_ it exists. So to prove "0.9... = 1" means to prove that the limit exists and equals 1.

So lets say we believe we have a proof that the limit equals 1 using an argument like this:

0.9... = x => 9.9... = 10x => 9 = 9x => 1 = x => 0.9... = 1

This is not a formal symbolic proof, but there is one thing we can see immediately: this proof does not make use of the completeness axiom. Therefore it should logically be the result of a sequence of logical steps starting from what I earlier referred to as the arithmetic axioms. Now here is the point where the surreals come in. The point with the surreals is that they contain rational numbers and the arithmetic axioms still apply. (To be clear, I haven't thought this through 100%, but I am almost certain this is true modulo my hand-wavy reference to "arithmetic axioms".) That means that that same proof should work inside the surreal numbers to also prove that the limit of the partial sums is 1. But here is the key important point. Within the surreal numbers, the limit of the partial sums is _not_ 1. So what does this tell us? The original supposition that there exists a proof only making use of the arithmetic axioms cannot be true. So the proof must make use of the completeness axiom. Well the surreal numbers is _not_ complete so that axiom doesn't exist there and therefore the fact that the proof exists and works within the real numbers and not the surreal numbers is not a contradiction.

Okay this is a bunch of logic mumbo jumbo and no grade student should be expected to worry about things at this level, but we can still give a more simplified version of the proof that actually is in essence correct. Some people in this thread use the argument: "Well since we know the number must be less than or equal to 1 and we also know it must be bigger than any number smaller than 1, then it must be 1." That argument (while a priori assuming convergence) is at least implicitly using an intuitive idea of completeness. Is it logically formal and 100% rigorous? Of course not. But it is explicitly making use of a property of real numbers while the first proof is not. I think if students are to be taught anything about the real numbers, then they should get some general intuition for this property. The algebraic version of the proof is invalid (my claim, which hopefully is at least a little more well-supported given this comment here) and it doesn't give the intuition they should (hopefully) get anyway.

In any case, hopefully this does some to help clear up what I've meant throughout these posts.

P.S. Finally, in response to this:

> By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."

So what? Are you saying those statements are contradictory? If so, how? And if not, what are you saying?

Sorry, I misread the blog post’s statement about infinitesimals, I thought it said surreals.

How did Euler actually write his proof, do you know, or have a link? I’ve poked around online but can’t find it.

I usually hear people say that he wrote the algebraic one (i.e. the one that I'm saying is invalid). But I also have no idea where he supposedly wrote it so I can't verify it. For me it's more hearsay.

By the way I wouldn't hold it against him if he did write the proof that way. I'm pretty certain all the logic/model theory that comes into play came long after his death. The surreal numbers certainly did.