And yet for some reason, you call proofs, assumptions.
Care to explain that?
There are in fact many different possible formal constructions, which are equivalent in the classical context but which come apart in other contexts (most notably, within “constructive”/“intuitionistic” mathematics, as in topos theory).
One quite natural and useful way is by saying that the real numbers are Dedekind cuts of rationals, where a Dedekind cut consists of a partition of the rationals into a downwards closed set L with no maximum element, a set with at most one element M, and an upwards closed set R with no minimum element.
The idea is that this represents how the real number is in comparison to every rational: it’s above those in L, equal to those in M, and below those in R. The “no maximum/no minimum” element rules ensure it is not infinitesimally close to any rational without being equal to it.
(One then defines arithmetic operations on the reals accordingly.)
Note that this has, as a consequence, that any two reals that are above and below the same rationals are equal. And, accordingly, any two reals that are infinitesimally close are equal.
We could present the same system via different axioms, taking that more explicitly as one of the defining rules. It’s just semantics what you call the foundational axioms and what you call the consequences.
He is a math expert. Despite the fact that I can’t find specifics I’m sure what he’s saying is right. Not just because he’s answered other questions in the past in great detail, but because of what I know from various mathematics courses. You can define a hell of a lot of stuff that is incredibly bizarre. Such as number systems with non-transitive or start-point dependent inequalities. Sorting the numbers on a clock in clockwise order starting at an arbitrary point is one such example where weird mathematical systems come into play.
Trust me, if 1 > 12 can be true depending on your rule system, there’s a lot that depends on your rule system.
I thought it might have to do with the Dedekind cut construction, but I didn’t have enough time as of that post to really digest the set construction notation fully. Thanks.
Based on what I know, he’s been flunking out. So knowing some set theory doesn’t impress me.
And I’m still waiting for an answer to my last question.
Did you miss his post a few hundred posts back demonstrating how in the hyperreals .999… doesn’t necessarily equal 1? If not, I suggest you go back and find it. I doesn’t answer your question directly, but it certainly shows he’s correct.
He’s certainly NOT correct about the proof I presented being based on assumptions.
Fair observations… and really inconsequestial to what I am now arguing, just for fun.
It has been stated that there is absolutely no end or that at least you could never get to the end of an infinite decimal expansion. I blieve i have refuted that. Have way at it. I am inviting it.
Okay, there are multiple number systems. The Reals and the Hyperreals are two of them, starting with different axioms. If he shows that the hyperreals, a system with different axioms, can have .999=/=1, then it follows, logically, that .999=1 is dependent on the axioms you can use.
Again, if I can construct a system (a system you use every day) where 12+1 = 1*, I’d say it’s pretty much a given that anything you care to present is based on axioms. Hell, the .999… = 1 vs .999 =/= 1 problem doesn’t even exist in the rationals, integers, or naturals. So you could say that some of the axioms in question are the very properties that make the reals and the rationals different sets.
- Or 23+1=0, for you 24-hour clock users.
And everything you just said is completely irrelevant to the issue at hand.
No, it’s not. You asked how the proof was based off assumptions. I pointed out that the assumptions are tied to the fact that you’re using real numbers. If there are number systems other than reals, and more importantly, systems other than reals where the proof fails then it follows that assumptions are key to the proof working, because they fail in cases where those assumptions aren’t present (i.e. not using Reals).
No, I know how proofs are based on axioms. I want to know why Indistinguishable is calling proofs assumptions. I think that’s a pretty egregious error and if you don’t, then I don’t know what to say to you.
Well, of course the definition is arbitrary, but it’s the agreed upon definition of the real numbers. If you want to impose your own definitions, you can make any relation between the symbols ‘0.999…’ and ‘1’ true: that 0.999… is smaller, larger, greener, more to the left, whatever you can think of; in this way, the discussion is pointless. And there’s an important difference between the reals and the natural numbers (though, of course, only if we use ‘reals’ and ‘natural numbers’ to mean what mathematicians typically take them to mean; again, you may redefine according to your tastes, which however makes meaningful discourse impossible), which is that there are more reals than natural numbers.
He never called proofs assumptions. He said that, specifically, in that proof, lim(n>inf) { 1/10[sup]n[/sup] } = 0 (or simply 1/10^{infinity} = 0) is an assumption. That’s a completely different statement.
It doesn’t matter if that assumption is also proven somewhere else. Within that proof itself that particular premise acts as an assumption of the way limits work.
To use a parallel, using the quadratic formula to find the roots of a parabola is an assumption of the quadratic formula. It doesn’t matter if it works, it doesn’t matter if you can prove it by completing the square on an arbitrary quadratic equation. If you do not prove it within the problem itself, it’s an assumption.
That’s what he was saying, that limits and the properties of infinity – regardless of their provability, axiomatic basis, whatever in the reals – were assumed within the proof you gave. The only thing that proof proves is that .999…=1, everything other bit of math used in that proof to reach that conclusion is an assumption.
Please define the term “.000…1”
You’re making some lively assertions, but you haven’t given us any definitions.
It is in no way an assumption. It flows directly from the definition of the infinitesimal calculus. To even attempt to call it an assumption is a gross insult to intelligence of your audience. I suppose you’ve been getting away with it up until now - which is surprising. I though that we had some actual scientists and mathematicians here.
You do realize that even 1+1=2 is an assumption about the properties of the “+” and “=” operators, right?
Those are axioms. Please use the correct terminology.
The behavior of limits is not axiomatic. It is only based on axioms. But I’ll tell you a secret. In deductive systems, EVERYTHING is based on an axiom.
Shhhhhh
Your points are valid.
I guess I was merely trying to express that perhaps, we impose this restrisction unecccessarly. I mean if you are using limits to define your numebrs, then there is no other way around it. There would have to be a real number between the 2, agreed. But I at the time I was trying to say we don’t have define numebrs this way.
Anyway… we make rules, but they are not (or at least should not) be completely arbitrary in math. I have a nagginng feeling limit definitons are imposed on numbers because people don’t like infinitesimals. Whish is ok, probaably makes sense too. But it also to me sweeps some issue under the rug. Unless you just want to say they plain don’t exist in any real way to begin with.
If you want to say numbers don’t exist in any real way but that which we imagine, I am sorry but I am not in that camp. I take a platonic view there.
I am more concerned now with whther we can or cannot “reach” or “find” the end of an infinite sequnece of 9s or whatever. Which I have been arguing for some pages now… With various bewlidering resonses in mind me anyway 
an infinite string of zero’s except for the 1/infinity decimal place. It seems a fariously obvious definiton [given my context], but i guess there could be some confusion.
The whole point of my “Everything in math is just rules” thing was to say “Everything depends on the definitions; with different interpretations, all this ‘0.999… has infinitely many 9s and is just less than 1, by an infinitesimal amount’ stuff is perfectly sensible”.
So, yes. Certain things flow directly from the standard definitions of the infinitesimal calculus.
And other things would flow with different definitions, or different interpretations of the terminology, or different axioms or different rules (these all amount to the same thing). That’s what I was saying. The question “Does 0.999… = 1?” is dependent on your interpretation of the notation. I’ve noted the standard interpretation, and also noted other, reasonable interpretations one might have used instead.
For example, you will find that Keisler’s textbook on calculus discusses the infinitesimal calculus in the context of different definitions (or different interpretations of the terminology, or different rules, or whatever you want to call it), such that 1/10[sup]n[/sup] is positive rather than zero when n is infinite. And why not? There’s nothing wrong with that.
Nothing in math is a brute external fact independent of our made-up rules, any more than bishops moving diagonally is a brute external fact independent of our say-so. That was my only point.
Well, that’s my point… When I said “Everything in math is just rules”, you went off on me. But now you are claiming essentially the same thing, yet taking it as somehow against me?
[One might terminologically quibble about some rules being called “axioms” and others being called “rules of inference” or “rules of logic”, but this is only a difference in our attitude; it’s all the same thing, whether we call them “rules”, “axioms”, “laws”, “definitions”, or what have you… The “rules of logic” are as malleable as any other; see classical vs. intuitionistic (vs. paraconsistent vs. linear vs. quantum vs. noncommutative vs…) logic]
You claimed
There’s no sense in saying “it’s not a rule but in actual fact…” about anything in mathematics, and particularly not about something which has been demonstrated to be quite dependent on the ambient rule-system, in the context of an argument about that choice of rule system.