Is 1 = 0.999... ? Really?

What does the concept of infinite refer to?

Endlessness, such as space extending without bound or the number system having no greatest number.

… or a repeating decimal that never reaches its resolution limit. :sunglasses:

How do you observe endlessness?
How do you observe an absence of an end?
How do you observe absence in general?
Finally, what is an end?

Basically, absence cannot be observed. Absence is an unmet expectation. When we expect to observe X but end up observing Y, we say that X is absent and Y is present.

Endlessness, being an absence of end, is an unmet expectation of an end. It’s not something you observe. It’s something you expect to observe but do not observe.

There is no endless process.

There are no tireless minions endlessly calculating one digit after another.

Both processes and algorithms require sequential steps and therefore time to complete. Since there is no time involved, it’s impossible for there to be any sort of process or algorithm.

I don’t need to observe the far side of the Moon to know that it is there. As I said, Logic dictates.

Neither he nor I was talking about taking time. We were briefly discussing the implied process of converting a ratio into a decimal as a “process”. He merely mentioned that it should be seen as a “completed process” and I mentioned that it is an “ENDLESS process”, thus never “completed” regardless of any time requirements.

Prediction refers to something that has been observed in the past that is expected to repeat in the future.

Everything has to be grounded in the observed. Otherwise, it is meaningless.

And such is the case with the concept of infinite. It is strictly speaking meaningless.

IOW, you redefined what the word ‘process’ means so that it suits your argument.

Okey-dokey.

Not necessarily, but even so, an infinite series is a prediction from prior observation.

Actually everything has to be grounded in Logic, else literally meaningless. Meaning stems from subtle unconscious logic. One cannot even observe until certain aspects of logic are already operational.

Not to those who understand how reasoning works. When you observe that you can always add another number to whatever number you have, you can “predict” that there is no greatest number = “infinite”.

No one redefined, nor defined anything. Come up with a valid argument or follow Carleas.

Absolutely necessary. Any prediction that does not refer to something that has been observed in the past is meaningless.

That’s very backwards. Logic precedes sensations? Really?

That makes sense. But only because you think it does.

Carleas left because he got sick and tired of your nonsense.

Yeah, I think it’s time to leave.

I am afraid this paradoxical forum harangues with the adage 'You can check in, but you can’t check out, mate.

You can only put up with someone saying “black is white” for so long. And this isn’t the only thread where it has happened.

Bullshit. He couldn’t respond to my last question:

Any time I ask a stumping question of Carleas, he either ignores it or simply disappears. He can never admit defeat (as most on this forum).

You guys continually make up distracting excuses, “you are redefining”, “you are ignoring”, “you are…” when the real truth is that your arguments are simply flawed. Whereas I need no excuses because what I state is coherent and logically valid.

Specifically, I lost interest after this post, in which James claimed that we can “only speculate” that (P) is the domain of a function (f:, P \to N).

Between that and refusing to acknowledge the distinction between a set that is a subset of a set, and a set that is an element of a set, I got the impression that continuing the conversation would mean explaining why tautologies are true to someone who wouldn’t admit it even if I could convince him.

Nah, I’m good.

Quite the opposite.

You try to inject irrelevant arguments to distract from the points being addressed. You quibble over what this or that word might mean to a mathematician versus common usage when those words hardly dictate anything relevant anyway. Meanwhile, you ignore the relevant arguments.

And it is certainly your speculation that your function covers the P domain. That is the whole crux of the argument. Oh but you just can’t tolerate it being questioned. Geeezzz…

…just more excuses to divert from the fact that your bijection function has a serious problem that I finally managed to sufficiently expose. You haven’t provided a valid bijection function and you can’t because N cannot pair with N plus A.

To be perfectly honest with everyone in this thread, nobody on earth actually knows why some rational numbers numbers do or don’t terminate such that this question even arises. Doesn’t happen with multiplication. This whole thread and all of these proofs are nonsense until someone solves that.

Nonsense!

Take 10/3 = 3.333… ← We know that 10 divided by 3 gives us 3 and a little bit. No mystery there. What is that little bit? It’s 1/3. Dividing 1 by 3 is the same as dividing 10 by 3 except an order of magnitude lower. So we should get similar results: 1/3 = 0.3 and a little bit? How much is that little bit? It’s 0.1… and it continues. ← There’s no mystery to this. We know why we get 3.333… when we divide 10 by 3. It’s just a pattern that reinforces itself on every magnitude lower.

You’re misunderstanding.

When you multiply whole numbers, you never get a number like: 64.786…

When you divide numbers, sometimes the answer is something like: 0.5 or 1.3…

0.5 terminates, 1.3… terminates into infinite regress (keeping it rational)

There’s an even finer point to this mystery that is often overlooked. When you place a bar over the 3 to signify its rational repetition, the threes actually regress, technically, at no point are those 3’s actually repeating!!

You seem to think that just because you have learned how to divide into decimals that you actually understand why some numbers terminate and others infinitely regress. I assure you, you have no clue.

You can multiply any two natural numbers but you cannot divide any two natural numbers.

Operations have limits. They cannot operate on any kind of input you feed into them.

You cannot, for example, take (3) apples and divide them in (5) sets of equal number of apples. (\frac{3}{5}) is an impossible operation. But you can say that this impossible operation is equal to some other impossible operation such as (\frac{6}{10}).

When we say (\frac{3}{5}=0.6) that’s all we’re saying: that (\frac{3}{5}=\frac{6}{10}).

Decimal number is simply a fraction of the form (\frac{a}{10^n}).

(\frac{1}{3}) gives you a non-terminating decimal when you try to represent it in decimal form because there is no natural number that you can multiply by (3) and get some power of ten.

There is absolutely nothing mysterious here.