Magnus Anderson wrote:Silhouette wrote:You're resorting to nonsense about conceptual matters being separate from empirical matters because deep down you recognise the only "traction" you have left on this issue is to insist that your argument doesn't "need" to be testable.

What makes you think that only empirical claims are testable?

Nothing, because I explicitly said that "Most of my objections are actually logical ones, and occasionally I back them up with empirical objections - either way, in order for you to forward your conceptual propositions as knowledge, they need to be testable and tested."

I never said

only empirical claims are testable - you are making things up AGAIN.

Magnus Anderson wrote:I never said nor implied that this thread is about untestable claims. I merely said that it's not about the kind of questions that you (and Gib) have been obsessing over ever since you joined this discussion.

This thread has nothing to do with science, epistemology and information theory. It has to do with mathematics and logic.

Unfortunately you don't even realise that you have been implying that this thread is about (your) untestable claims at

multiple points. Every time you claim you don't "need" to prove your (il)logical claims, or that the logical arguments of others either don't exist when they clearly do, or you simply

assert they're false when you notice them whilst at the same time accusing others of that very same fallacy that you're committing,

as well as every time empirical tests are offered to help you understand your errors that you simply claim are irrelevant - you're arbitrarily invalidating the possibility for both logical and the empirical disproofs to apply to your attempted "proofs".

Basically you're constructing epistemological double standards for this thread to make it impossible for validity to prove you wrong whilst simultaneously dictating that invalidity can prove you right. In itself this is fallacious - "moving the goalposts". This is why you have no intellectual honesty, integrity or authenticity.

Homework for you: devise an argument that would successfully cause you to change your mind that isn't simply a reiteration of your own points and how you think they're right.

That is intellectual honesty - the ability to evaluate both sides of the story: the existence of which I've been trying to communicate to you all this time.

Obviously you've not been in higher education, or you'd be versed in this kind of thinking because that's what qualifies you for it and qualifies you to succeed in it. Either that or you completely disregard this kind of thinking especially for the internet - but from what I've seen I'm going to have to guess you never even made it, and perhaps realised it was never worth you trying. Must be the fault of the institutions though, right? Never yours.

Magnus Anderson wrote:Silhouette wrote:Why does B' now have an extra 0?!

Because that's what you're doing. You're creating a new set, \(B'\), by adding \(0\) to it (the extra \(0\) you speak of) and then by adding

all of the elements of the set \(A\) while relabelling them in order to make the resulting set \(B'\) look identical to \(B\).

\(B = \{0, 1, 2, 3, \dotso\}\)

\(B' = A \cup \{0_{new}\} = \{0_{new}, 0, 1, 2, 3, \dotso\}\)

\(B' = \{0, 1, 2, 3, 4, \dotso\}\)

In the final step, you're renaming \(0_{new}\) to \(0\), \(0\) to \(1\), \(1\) to \(2\), \(2\) to \(3\) and so on. That's how you create the illusion of one-to-one correspondence between the two sets.

You're correct that I intentionally add an extra \(0\) to \(B\).

You're incorrect that I'm "relabelling" the elements of set \(A\). I'm adding \(1\) to each of their elements and appending the result to the \(0\) that I first append to \(B'\). The fact that this makes them identical to \(B\) is no accident - this was intentional to show you that alternative set construction to end up with the same infinite set doesn't objectively change the set even if it now "looks like"

injection when in fact set identity ensures

bijection.

Obviously my efforts were yet again wasted.

Magnus Anderson wrote:Silhouette wrote:Either way this simplifies as \(B' = \{0, 1, 2, 3, \dotso\}\), which is identical to both B and A as you accepted from the very start...

It does not simplify. When you write it like \(B' = \{0_{new}, 0, 1, 2, 3, \dotso\}\), it's pretty obvious that \(B'\) is greater than \(B\) because it has all of the elements that \(B\) has plus the new zero you added. But when you "simplify" it, you make it look as if they are identical. That's the trick.

So where's the extra element in \(B'\)? Well, \(0_{new}\) is the extra element. Don't conceal it and you'll see it.

I use the term "simplify" in the technical sense of making absolutely no change to the essence of the contents, not colloquially such that it potentially implies concealment or trickery. My terminology is wasted on you.

\(B'\) has all the elements that \(A\) has, with 1 added to all of them before they populate \(B'\) - to the preliminary ends of \(\{1, 2, 3, \dotso\}\)

That's why I add \(0\) first so that this set starting from 1 is appended to a set that has \(0\) already prepopulated within it.

I defined \(B'\) as \(\{0, 1, 2, 3, \dotso\}\) - stop pretending I didn't and actually read my explanation. No trick.

The only "trick" is if you mistake bijection for injection because of the intentionally alternative appearance of \(B'\)'s construction compared to \(B\) (and \(A\)) to conclude that \(B'\) now has more elements when if you follow each corresponding element to infinity it's literally impossible to find any injection.

Magnus Anderson wrote:Silhouette wrote:I think you're done, don't you?

That would be you.

I wish it could be, but unfortunately I'm waiting on you. I won't give up until you've learned something: you the self-confessed non-mathematician professing mathematical expertise.

What would you most like to take from this thread? Victory? Self-improvement? Reconciliation? Teaching your expertise? Name it.

Magnus Anderson wrote:Silhouette wrote:Every partial product is greater than zero, yes - because it's only partial! Thus the same logic carried over to any "non-partial-product" of some kind of "infinite product" is invalid. Again - poor logic skills on your part.

Every partial product of \(0 \times 0 \times 0 \times \cdots\) is equal to \(0\) and that's precisely why its result is equal to \(0\).

The same does not apply to \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots\).

I can repeat myself too if you want.

That's not getting us anywhere though is it.

I know every

partial product is greater than 0. This just panders to your appeal to finitude that you keep denying you're doing.

INFINITES. We're dealing with them, not "partial, finitely somewhere along the way of an infinite series" which you're pretending still counts as valid for evaluating an infinite series.

Magnus Anderson wrote:Silhouette wrote:The fact that it's impossible to narrow down any possible value should ring alarm bells that this "proof that the gap exists" is flawed.

Who told you it's impossible? I said it's unnecessary.

I showed you that it's impossible to narrow down any possible value above 0 to fill in any "gap" between \(1\) and \(0.\dot9\)

The only value that cannot be divided smaller, thus satisfying this "gap" is 0, otherwise it can always be made smaller.

I'm

showing you that it's literally impossible. You're just saying it's unnecessary for it to be possible, but you're calling it "unnecessary to prove an exact amount".

The problem is that you don't appreciate the implications of your vagueness.

This is mathematics, not speculation.

Magnus Anderson wrote:Silhouette wrote:The reason we care about the poor guy

It's because you're a confused and at the same time conceited mind (hence your defensiveness.)

You're picking up that I'm defending logic and reason huh? Sharp!

Defending

myself though? Fuck that - I don't give a shit what anyone thinks of me unless it suffices as a means to the end of enabling the grander scheme of rationality to be upheld. I don't want that goal to elevate me or bring me down, it has nothing to do with me - rationality is objective and has nothing to do with the person who forwards it. It works regardless, independently. The only personal satisfaction that I take from philosophical discussion is in the aesthetics of truth. It's nice. Obviously you don't think so, but that's a result of your personal issues and has nothing to do with me personally. Call this kind of dissociation from conceit "conceit" all you like. I give zero shits.

Magnus Anderson wrote:Silhouette wrote:I mean, it literally necessarily is the case by definition - you even accepted the fact that they are identical sets insofar as they have one-to-one correspondence.

I accepted that \(A\) and \(B\) are identical, not that \(B\) and \(B'\) are identical.

\(B'\) is defined to biject perfectly with \(B\), just with alternative construction to make it "look like" it's different to people fooled by the superficial.

As I keep saying and proving, that's you.

Magnus Anderson wrote:Silhouette wrote:Take a break. Relax. Mull it all over - it'll be good for you.

Yes, it will be good for you. If anything, you'll be less focused on who you are in relation to other people and more focused on the actual subject.

Can you do this for us, please?

Or do you REALLY have to tell us how great you are each time someone says something negative about you?

I'm not giving up on you, Max.

No matter how frustrated it makes me.

I will make you learn even though I've been taking the entirely wrong approach of being honest with you about yourself this entire time. I should have been softer and less confrontational - I already knew that this just makes people act irrationally in exactly the same way that you are. I did the same thing with

Urwrongx1000 but after a couple of years even he realised I was onto something.

I'm willing to wait that long for you, Magnus.

Magnus Anderson wrote:\(0.\dot01\) is a number that is greater than \(0\) but smaller than every number of the form \(\frac{1}{10^n}\) where \(n \in N\).

Oof! If you meant N as \(\Bbb{N}\) as the set of natural numbers you're in danger of starting to sound like you've been studying up on the subject!

Now we just need to eliminate your reliance on the invalid term \(0.\dot0{1}\) and maybe we can get somewhere?!

Perhaps if we formulate it like "Given the quantity \(x\) such that \(x=1-0.\dot9\), \(0<x<\frac1{10^n},\forall{n}\) where \(n\in\Bbb{N}\)" we might start sounding remotely close to professional.

But I'm just a mathematician, I'm not a professional one, so if anyone wants to correct me with any legitimacy - please feel free. I'm pretty sure there's a more succinct and correct, purely symbolic way of legitimately expressing that.

But this is just notation - the argument remains that whilst \(\forall{n}\) where \(n\in\Bbb{N}\), \(\frac1{10^n}-0\gt0\), \(\prod_{x=1}^\infty\frac1{10_x}\not\gt0\)

Basically "partial" products don't cut it by definition - like we already covered.

Magnus Anderson wrote:That's what \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots\) means. It is telling you "This is a number greater than \(0\) but less than \(0.1\) and \(0.01\) and \(0.001\) and \(0.0001\) and so on".

You can say "But there is no real number that can represent such a quantity!" to which I can respond with "But the set of real numbers is not the set of all quantities." I can then add that numbers are merely symbols invented by humans in order to represent quantities that are of relevance to them. They are not quantities themselves. Symbols are few and far between, quantities are numerous. And when it comes to quantities, they can be as small or as big as you want them to be. There is no limit.

All in all, all you have to do is understand what that infinite product represents. But you're refusing to do so, obviously because you have your own agenda -- it's in your interest to misinterpret it.

What's certain is that \(0.\dot01\) is not a number that is less than itself. That's your own invention.

I know what the expanded product of your infinite product is intended to represent.

I know that "there is no real number that can represent such a quantity" - as you say.

It is therefore your duty to comprehensively a valid population of quantities that satisfy the ability to exist between \(0\) and the difference between \(1-0.\dot9\)

You need to stop cowering from the formalisation of such a proof, apologising that it's "not necessary" by virtue that it "ought" to follow from what you regard to be a valid argument, which I have proven to be logically invalid.

You need to do better.

The Hyperreals already attempted this way before you, but as I've argued their conflation of the infinite with the finite is only a useful tool rather than truth.

It's not enough to say that "there

should be some quantity there", you need to validly justify it - my "agenda" is with intellectual rigor ONLY. I don't give a rat's ass about anyone's excuses to get around that - I will hunt down your mistake and destroy it. It's fun and I enjoy the aesthetics of the result, but I don't care one jot what form it ultimately takes - even if it turns out you're ultimately right somehow.

It's a fact that \(0.\dot0{1}\) is a contradiction regardless of the fact that it was me who identified it. You cannot logically declare an

infinite series of zeroes entirely bounded by a decimal point and a \(1\) because entirely bounded contradicts the boundless quantity of zeroes in between. Simple. Logic.