Magnus Anderson wrote:Silhouette wrote:Let me show you how hypocritical this example still is:

I suppose you mean contradictory.

Idiotic would be better, actually.

Pick one, they're all entirely appropriate

even if you pretend only one (your one) is correct - oh wait, that's what you've been doing this whole time - at least I'll know what stupidity to expect.

Magnus Anderson wrote:Silhouette wrote:Your objection to \(\frac{9.\dot9}{10}=0.\dot9\) is that at some "end" to the infinite recursion of 9s there's a spare 9 for \(0.\dot9\) that doesn't match to \(9.\dot9\)

My objection is that if \(0.\dot9\) is a symbol representing the same infinite sum with the same infinite number of non-zero terms wherever it appears, it follows that \(9 + 0.\dot9\) is an infinite sum that has one non-zero term more than \(0.\dot9\). This is because it has all of the terms that \(0.\dot9\) does plus one more. And when you divide it by \(10\), the number of terms is preserved, so the resulting number, even though similar in appearance, isn't really equal to \(0.\dot9\).

On the other hand, I have no idea what it means to say that at some end of the infinite sequence of 9's there's a spare 9. You'd have to clarify that.

Are you

trying to ignore the connection between \(\frac{9 + 0.\dot9}{10}\) "having 1 more term" than \(0.\dot9\) and having "no idea what it means to say that at some end of the infinite sequence of 9's there's a spare 9"?

Or are you

unable to see it? It's one or the other.

\(\frac{9 + 0.\dot9}{10}=0.\dot9\)

So if it has "1 more term" than some other \(0.\dot9\) then where is it?

It's not in front of the decimal place.

The 9s after the decimal place go on forever

with no end. With these nonsense notions of \(0.\dot{0}1\) that you tried to conjure out of your ass before, you were pretending you could fix some term at the end of an endless sequence, so maybe you're proposing this "spare 9" performs that logically contradictory feat? Nothing else works for you, so where are you going to retreat to now?

\(0.\dot9\) certainly has one-to-one correspondence with \(0.\dot9\)

Again, your logic works

only for finitude - not infinitude.

Magnus Anderson wrote:Silhouette wrote:Yet somehow for your visual display of \(0.\dot9\)s, starting one "1 decimal place" after the other makes it shorter. Here the ends are equal, but previously they weren't.

What "ends" are equal? And what "ends" were previously not equal?

In your visual display you start the purple \(0.\dot9\) after the green by shoving another yellow term in front of it that's not even part of the purple set that you've pushed over anyway.

But even if the purple could be justified as starting after the green, it would only therefore be shorter if the ends of each (infinite) set "ended" at the same point.

But as I just explained above, your previous example tries to have two sets of \(0.\dot9\) "constructed in different ways" starting at the same point, but one "had an extra 9 added in at the beginning before all elements were shifted over" therefore each (infinite) set would have to "end" at different positions for one to be longer than the other.

Inconsistency. Arbitrary. Contradictory. Hypocritical of you to accuse these things of me.

Magnus Anderson wrote:Silhouette wrote:You can arbitrarily match one-to-one correspondence however you like so that A can be any "size" larger or smaller than B and vice versa.

You can arbitrarily determine how elements of one set are represented in the other set. But once you do so, you can't arbitrarily decide whether the two sets are equal in size or not.

I can say that odd numbers in \(A = \{1, 2, 3, \dotso\}\) are represented in the following way in \(B = \{1, 2, 3, \dotso\}\):

$$

1 \mapsto 1\\

3 \mapsto 2\\

5 \mapsto 3\\

\cdots

$$

This means that \(1\) in \(A\) is represeted as \(1\) in \(B\), and \(3\) in \(A\) is represented as \(2\) in \(B\), and \(5\) in \(A\) is represented as \(3\) in \(B\) and so on.

Once you accept this, you can't say that two sets are equal in size. They aren't. Every member of \(B\) is a member of \(A\) but the reverse isn't true.

But you are not listening, so you keep making one mistake after another and getting frustrated.

Listening to you making one mistake after another and insisting you're right is what's getting me frustrated!

You literally created identical sets, i.e. bijective by definition, \(A = \{1, 2, 3, \dotso\}\) and \(B = \{1, 2, 3, \dotso\}\), and arbitrarily matched them with the appearance of one-to-one correspondence such that only the odd numbers in A are counted, and therefore concluding that A is a different size to its

identical counterpart, B.

I'll give you the benefit of the doubt and assume you meant the set, A (with only odd numbers) was being compared with B such that \(A = \{1, 3, 5, \dotso\}\) rather than \(A = \{1, 2, 3, \dotso\}\).

Even then the one-to-one correspondence is with the natural numbers, x in B with 2x-1 in the original A.

You're wrong either way, and I'm right either way.

If you took just a minute or two out of your busy life of complacent assertion and read up on bijection, you would know that the function \(f: R\mapsto{R}, f(x) = 2x - 1\)

is bijective.

Magnus Anderson wrote:It's not arbitrary. What you're doing is arbitrary. You're the one parting ways with logic.

You're the one misunderstanding quantity and getting fooled by decimal notation such that you think that one-to-one correspondence between the first element in one set and the first element in another set is "wrong" because you put them in different columns in decimal notation.

This parts with logic on superficial arbitrary grounds.

Magnus Anderson wrote:Silhouette wrote:Convenient how you can judge sizes arbitrarily to fit your point, no?

That's precisely what you're doing.

Yes it's so arbitrary of me to suggest one-to-one correspondence of first elements in a set rather than cooking up some pseudo-logic to match the first with the second just to give the illusion of different "size" - size of infinites no less! They all go on forever, some don't go on "more forever" than others as I've said so many times and you've still not accepted.

Magnus Anderson wrote:You just contradicted yourself. You said you can't define the word "infinite" and then you went on to define it by saying it means boundless.

No, you just conflated signifiers and with signifieds to think that was a contradiction.

Please tell me you think I made up that terminology so I can annihilate you all over again in yet another way.

All

words obviously have definitions, but the things they refer to can defy definition - making the act definition questionable in the first place. "The defiance of definition" can suffice as a definition of a word in reference to some undefined aspect of existence. The signified has infinitude, but the signifier implies at least some finitude for it to be a word at all. There's obvious problems with the truth in doing this, but undeniable utility in doing so - hence why people perform this questionable act in the first place.

Only in this way with these concessions can the word infinity have definition and even synonyms like boundless, which is only a "definition" in the same way that a tautology gives extra information (it doesn't). There's only an appearance of definition here (which again is your whole problem), and on top of that

infinitude is an absence of finitude (definability) rather than a definable thing itself. Saying that which is infinite is provably definite is like asking someone to prove the nonexistence of absence - as well as being a logical contradiction.

There's always so many things wrong with everything you say!!!

Magnus Anderson wrote:Yes, the word "infinite" means "boundless" but it does not mean "boundless in every way one can think of". It means "boundless in some ways" where "some ways" can be "one way", "two ways", "three ways" or "all ways". Yes, it can mean "bounded in all ways" but not necessarily. Its exact meaning depends on the context.

When people speak of infinite sets, they are not talking about all-encompassing sets i.e. sets that contain literally everything there is, they are talking about sets that have an infinite/endless quantity of members. That's why it's not a contradiction in terms to speak of infinitely many things happening within a finite period of time.

This is already how I ascertained you were thinking of infinite

a week ago and here you are again thinking you're telling me something I don't already know/understand. Again.

I criticised the use of "infinite" in reference to sets when any aspect of their construction is a product of finitude. I mentioned how even the natural numbers have a finite starting point on the number line, never mind the line being finitely bounded in all other dimensions as well, being infinite in only one dimension in one direction. It's finite in many more ways than it is infinite, yet it's still called infinite because it's infinite in at least one way. I also explained that even in other sets infinity is still in only one way, just with less and less finite constraints the more "types" of numbers you add. In other words, any size of "infinite sets" is determined by their relative lack of finite constraints and not any "different size of infinity". It's only if you could remove all finite constraints to "infinite" sets, that you'd get an entirely infinite set, which

would mean "boundless in every way one can think of". But this would require complete consistency - not your strong point.

Instead, you're happy to refer to all sets that have infinity involved in them in with the exact same term, no matter how many more finite constraints that they also have "but just in a different context". Stay as vague as you can to maintain only the amateur appearance of validity, right?

If you had a basket with several oranges and one apple in it, would you call it a basket of oranges? Obviously not, but people like you will flatly call any set "infinite" no matter how many more ways it had finitude involved in it.

Magnus Anderson wrote:Yes, you are still telling me what to think.

I'm telling you TO think.

And I'm showing you how to do it logically.

If you do that then you'll end up thinking what I think and what the professionals think, but for you to accept and understand it, you have to do it yourself (think logically).

But you can't even seem to identify my explanations as explanations - simply asserting that they're just demands and pretending you're thereby unchallenged.

Clearly there's

something going wrong for you to object to thinking and doing it logically - but I can only lead a horse to water, I can't make you

drink think.

Magnus Anderson wrote:Silhouette wrote:And you actually think I'm being silly about the extensive proof that 1+1=2

Yes, you are being silly. We may need an extensive proof to acquire a "deep" understanding of why 1+1=2 but I wasn't talking about the "deep" understanding.

All in all, you're being pathetic.

I'm being utterly serious, and your "Argument from Incredulity" (if I cannot imagine how this could be true, therefore it must be false)

is pathetic.

I know you aren't talking about a

deep understanding!!!

Apparently you think a deep understanding of what we're talking about is pathetic.

Enough said.

obsrvr524 wrote:Silhouette wrote:And you actually think I'm being silly about the extensive proof that 1+1=2

Reference?

Principia Mathematica by Alfred North Whitehead and Bertrand Russell.

Building up the foundations of maths, they finally reach 1+1=2 by page 362 (in the 2nd addition at least, in the 1st addition it's page 379).

Goddamnit, why is it so impossible for amateurs to believe that someone else might know things they don't?

Dunning Kruger effect, obviously. But fuck me is it frustrating to have to deal with.