obsrvr524 wrote:Isn't it by definition that there is always a number smaller than 1-0.999...? Infinity isn't a destination. Infinite means there is always more. That "always more" is the exact difference between 1 and 0.999....

It is part of the very definition of 0.999... that it is not 1.0.

It's by definition that there's always another \(9\) in \(0.\dot9\) and therefore never ever any room for anything at all between \(1\) and \(0.\dot9\) no matter how hard you go.

"Infinite means there is always more" - sure: just more 9s in this case though. The recurring "dot" is applying to the 9 it's on top of, not anything else.

The result of doing this is that there's "infinitely no space left": an infinite lack of space for even one finite infinitesimal - never mind an infinite number of them!

The idea that dividing one by 3 and then multiplying the result by 3 leaves an infinite gap between what you started with and finished with... - this should be a pretty good clue that you're inverting the sense of what infinity is if you think that. There's no infinite gap when dividing 3 by 3 and then multiplying the result by 3 - be consistent.

The idea that the closest possible decimal quantity to \(\frac{1}3\) is \(0.\dot3\), therefore there's an infinite gap between them should sound some alarm bells too.

obsrvr524 wrote:I can answer that one for James. He clearly stated that in order to use maths with infinities, you first have to define a standard infinity. That's where his infA originated (as opposed to infB).

This is perfectly circular by James here.

To have the conditions to do something (use maths with infinities) you need to have those conditions (a defined infinity).

I can clearly state that in order for black to be white, you first have to define a black white.

The conclusion that the in-finite can be de-fined mathematically is assumed in the premise that a standard in-finite can be de-fined.

obsrvr524 wrote:And his reasoning was simply that trying to say "2 * infinity" is like saying "2 * length". For it to make any sense, you have to be more specific.

Again with the treatment of the quality of having no quantity as quantity.

"For non-specified ending to make any sense, you just have to be specific with the non-specified ending"...

Magnus Anderson wrote:Ecmandu and SIlhouette are telling us that even if you define a standard infinity,

I am saying that defining (giving ends) to infinity (endlessness) is a contradiction in terms.

Magnus Anderson wrote:you cannot add two infinities together to get a bigger infinity.

This is correct though.

Magnus Anderson wrote:The difference between \(1 - 0.\dot9\) can be represented using a number. James's \(0.000...1\) is a number. Indeed, the very expression \(1 - 0.\dot9\) can be considered a number in the same way that a fraction of two integers is considered a number (a rational number.)

I forgot to mention that numbers have to avoid contradictions in terms, how silly of me.

"0.000" defines the start and "1" defines the end. With all ends accounted for, the "..." attempting to imply infinity is a contradiction in terms.

This is what happens when you try to define infinity by giving it ends: you make it finite (the opposite of what it is).

If there could be an infinity of something, e.g. 0 digits entirely enclosed by finite bounds, then the "finite" bounds would be indefinitely separated such that they'd never come to exist - just like the end of infinity never comes to exist by both definition and derivation.

\(0.000...1\) is a numerical representation of an invalid number. Just goes to show that just because you can represent something numerically, doesn't mean it's a number. Fortunately my logic went in the other direction, that to be a number, you have to be able to represent it numerically. Let's avoid "affirming the consequent" here, eh?

Magnus Anderson wrote:What you cannot do is you cannot represent the difference using a decimal number. And that's exactly what your so-called argument against the existence of numbers smaller than \(0.000...1\) boils down to. You're basically saying that if a quantity cannot be represented as a decimal number, it does not exist. But that's simply not true. "Decimal number" is not synonymous with "number".

So as I was saying, numbers are necessary in any

valid format. The invalid \(0.000...1\) therefore doesn't count.

I'd accept any valid numerical representation, decimal or otherwise.

\(\sum_{n=1}^\infty\frac9{10^n}\) or \(0.\dot9\) are fine because here the infinity isn't finite nor the finite infinity. The finites and the infinites are separate: the finites are defined in a very specific way such that the infinity can then say "now go do that defined thing an infinite number of times in the one way that infinity can be infinite".

\(\frac1\infty\) is not fine because the infinity is standing in for a finite, or it is represented as a finite - they are not separated. Infinity alongside any numerical format of finites is like an operation, as Ecmandu said. It's an instruction to operate on defined finites in an ongoing way - it's never "the end" of having been operated on - because obviously "the end" i.e. finitude is the opposite of endlessness. \(2\times\infty\) is invalid for the same reason.

Just be consistent! That's all you need. Math wouldn't be math if it wasn't consistent.

Magnus Anderson wrote:Consider that there is no decimal representation of \(\frac{1}{3}\). Yet, \(\frac{1}{3}\) represents a real quantity. It's a quantity that can be represented as a rational number (\(\frac{1}{3}\)) as well as a base-3 number (\(0.1\)). The fact that there is no decimal representation of \(\frac{1}{3}\) does not mean it's not a representation of a quantity.

Again, this violates consistency.

Why is it okay to divide 9 by 3 to get 3 and then multiply that by 3 to get back to 9, but not ok to divide 10 by 3 and/or then multiply the result back by 2 to get \(10\) or \(9.\dot9\) by using the exact same method?

You'll no doubt accept that 10/4 is 0.25 but somehow as soon as the remainders don't resolve to 0, it's not valid?

You probably didn't even stop to think that your logic here would invalidate the irrational and transcendental numbers, right? I bet you think they're okay, but the exact same thing isn't okay when it goes against your point.

Magnus Anderson wrote:What makes you think that hyperreal numbers aren't numbers?

You can call black "white" and it would still be what it is and and not its opposite.

I keep saying it can be interesting to see what happens if you treat infinites as finites, but that doesn't make it

true - only

useful.

Only count numbers as numbers and you'll be up to speed.

Magnus Anderson wrote:If you're asking for a rigorous proof, I don't have one. And I don't think it's necessary.

But then this says it all really.

I'm trying to get you to sufficiently delve into the details but if rigor isn't necessary to you, there's not been any point in me trying.

Magnus Anderson wrote:You provided no compelling arguments, no proofs. You merely defined the word "infinity" in your own way and used the logical implications of such a concept to "prove" that I am wrong even though that's not the way I define the word "infinity". (In fact, that's not the standard definition of the word.)

Infinity is the quality of having no quantity is something I've said over and over, which is exactly how it's both defined and derived. This isn't "my own way", it is "the" way.

These kinds of resorts of yours are pretty low - to just deny that all this work people have gone to for your sake was never even done. It's at the very least rude, if not outright ignorant.

Magnus Anderson wrote:The "straw/box" argument merely shows that he does not understand the simple fact that there are no inherently correct definitions of words.

I'm sorry, I don't accept the definitions of the words you've used to say there are no inherently correct definitions of words.

You see how self-annihilating this kind of argument of yours can get?

If you can't legitimately define the terms for your argument to be valid, then throw out words altogether, right?

Try imagining the possibility that you're wrong about something and think about how you'd react and resolve it. If the answer is to double down indefinitely, then if neither of us ties up this resolved issue, this debate will go on endlessly. Defining a finite resolution to such an infinite scenario would be invalid - is invalidity what you're aiming for?