With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed. — Michael

I think that's a sensible direction.

But does that imply necessarily that time and or space in our universe must be discrete and not continuous?

or how many pizzas you can buy, if they cost 50 cents each and you have a dollar.

When it comes to the supertask of counting a "countably infinite set", by exponentially decreasing the time it takes to count, here's the problem I have with the scenario in the OP:

Tthe mathematical ability to count in a finite time is purely mathematical and, crucially, doesn't involve the concept of conscious experience.

Once you decide to make this supertask accomplishable by *a human mind*, then you run into brand new problems that don't exist in a purely mathematical context. Let me explain:

You say he halves his rest period every step - but it's still implicit that each step has to be a *conscious experience*, a "choice". Which means, even though you can mathematicaly say he can count to infinity in 1 minute, I propose that he would be stuck consciously in that 1 minute for eternity, since that 1 minute includes an infinite series of choices and conscious experiences.

So if you send a person down such a staircase, in some mathematically perfect platonic realm where time is infinitely divisible, that person will be stuck in the eternity of that 1 minute.

What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks — keystone

So why don't you just link me to the reading materials that would lead me to believe that the supertask you described in your op is possible to complete? That specific supertask, not supertasks in general. Let's not beat around the bush, let's get right to it.

Sadly, I am not good enough at maths and logic, so I can't post valid or interesting comments regarding this paradox. What I try to defend is that what keystone wrote is actually a paradox. — javi2541997

I think that if you're not good at maths and logic, I would think that you might not be in a good position to know if this is a valid paradox or just straightforward nonsense. If it is a paradox at all, it would only be in a mathematical sense. Surely not EVERYTHING contradictory is a paradox.

That comment doesn't justify why the person should reach the end of the stair case.

Here's what Wikipedia says about paradoxies

A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation.[1][2] It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.

`despite apparently valid reasoning from true or apparently true premises` - that's key! The premises and steps in reasoning have to make some kind of sense.

The premise that you can reach the end of an endless staircase doesn't apparently look like valid reasoning or true premises to me.

As far as I understand about paradoxes, that's precisely what a paradox is about. It is a self-contradictory statement, but arrest our attention. — javi2541997

Yes, but usually where the contradiction occurs exactly is supposed to be *non-obvious*. "He went down some endless stairs, and reached the end". It's not non-obvious where the contradiction is. It's immediately obvious.

Because it's obvious, it's not so much a paradox as it is just a plain contradiction.

This is a paradox I've come up with myself. But as Michael has mentioned it's very similar to Thomson's lamp. Where do you see problems with it? — keystone

I don't see a reason to think that a person will reach the bottom of the infinite staircase, ever. You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts.

"There was an old woman. She was only 2 years old, and really just a baby." There's my paradox.

Then where is the reaching the bottom in under 1 minute coming from? — Benj96

He just made it up, it doesn't come from anywhere. That's why I'm questioning if it's really a paradox, that's why I want another source for it

I don't think the intention was for physics to be a problem. It's probably supposed to be a purely mathematical problem, it's too fantastical for physics to be a concern.

where does relativity come in?

Is there another source for this paradox? Or did you just invent this yourself?

I get an occasional email from Quora about interesting questions and answers of the day. By pure serendipity this came up today:

https://www.quora.com/Whats-the-craziest-mathematical-fact-you-know/answer/Brian-Overland-1?ch=15&oid=164968393&share=3b2e848f&srid=o1wj&target_type=answer

The whole answer is related to this thread, but the second to last paragraph is of particular interest to me in regards to this theoretical dice:

(Which is one reason you cannot even in theory randomize across the natural numbers with uniform probability.)

This leads me to think that maybe, after all, it's not safe to assume god could prepare such a dice.

And here's a stack exchange conversation on why.

https://math.stackexchange.com/questions/14777/why-isnt-there-a-uniform-probability-distribution-over-the-positive-real-number

This is equivocation. There is "colour" as an object's surface disposition to reflect a certain wavelength of light and there is "colour" as the mental phenomenon — Michael

Yeah, this confusion seems common.

I'm not seeing what you're seeing in that comment.

Distal objects being part of the casual history of an experience doesn't make them constituents of the experience, any more than shovels are constituents of holes.

When I mentioned that the dice with infinite sides are fair, I was specifically referring to each side having an equal chance of being rolled. After all, God is fair. :P — keystone

Then why did you point me to your post where 1 takes up half the space on the die :P

The probability of Adam winning is exactly 0%. — keystone

How? You said 1 takes up a full half of the die - 1 is less than 1000, and the other guy has a 50% of rolling a 1. He has a 25% chance of rolling 2, so that's already up to a 75% chance to win now. Where are you getting 0 from?

Well your mistake is assuming it drops to 0%. If the die is split the way you said, it absolutely does NOT drop to 0% unless he rolled a 1.

You could only make a good argument that it drops to 0% if every number on the die had an equal probability - but the die you laid out does not have that feature.

I guess you mean this post. This doesn't seem like a paradox to me. Where do you think the paradox is?

that extra stipulation makes it a tautology, since a non hallucinatory experience of a distal object by definition requires the existence of a distal object.

So I have to agree with you, because I agree that tautologies are valid

also believe that distal objects are constituents of experience in the sense that you could not have an experience of a distal object without them. — Luke

What are hallucinations if not an experience of a distal object without a distal object?

It probably doesn't make sense to roll an infinite sided dice in the first place

I see, cool, thanks

so what was 30 years ago?

What, to you, is the difference between those two things?

but a question of whether or not we perceive the world directly. — Luke

I don't understand what work the word "directly" is doing in that sentence. Why not just say, "whether or not we perceive the world"? How does adding the word "directly" change the meaning of it?

I will give the best example I have: being (viz., ‘to be’, ‘existence’, ‘to exist’, etc.). When trying to define or describe being, it is impossible not to use it—and I don’t mean just in the sense of a linguistic limitation: it is impossible to give a conceptual account without presupposing its meaning in the first place. — Bob Ross

This occurred to me in the "I think therefore I am" conversation.

I think there are atomic ideas for sure.

I just think it's the next natural question lol. You said it was an example of knowledge without belief. There's nothing hostile or insulting or anti philosophical for me asking the next natural question, which is, do you believe this thing you said you know?

There's literally not an ounce of hostility in that. It's a pretty straight forward question, it's not a trick.

It's my understanding that the most basic definition of "belief" is just "something you take to be true."

So if the question is, "can you have knowledge without belief?", you can rephrase it as, "Can you know something if you don't take it to be true?"

Can you?

insofar as if I say yes, I believe a query has been made, than my knowledge of it appears predetermined and I’ve contradicted myself, and if I say no I don’t believe the query has been made leaves open the catastrophic descension into that pitiful sophism, you can’t know what you don’t believe. — Mww

First of all, do you beat your wife questions have the SAME implication if the answer is yes or no, not different implications. "Yes" means I do now and I used to. "No" means I don't now but I used to. Both answers imply "I used to beat my wife."

But my question I asked you doesn't have that. There's not any implication that remains constant with both answers.

And it's not an insult to philosophy, that's ridiculously melodramatic. It's the natural next question. You said you think knowledge doesn't require belief. I asked you for an example of knowledge without belief. You gave me the example of you knowing this, OF COURSE I'm going to ask you if you believe it. That's what happens next. That's just the script.

Instead of worrying about how much you dislike the consequences of the answer yes, or the answer no, why don't you just answer honestly? Avoiding asking an entirely philosophically pertinent question because you don't like the consequences of your answer is more of an insult to philosophy than anything else going on here. I don't want to insult you, I didn't intend for the conversation to be this hostile and I'm honestly quite flabbergasted that a completely innocuous question led to this, but if we're going to be throwing around melodramatic statements like "insult to philosophy", let's get real.

I don't understand why you're reacting like that. This kind of question is literally what is meant by the statement "belief is required for knowledge". There's no reason for you to get testy about it or say it's like "do you still beat your wife?" It's nothing like that. Either you believe it or you don't.

It's not an insult. I'm not insulting you. Are you perceiving it like an insult?

Again, "knowledge requires belief", as a statement, literally means if you know x, you must believe x. There's nothing insulting about me asking you that, it's entirely pertinent to the question and I would really like it if you could talk to me without getting heated about this entirely inoffensive question.

you said you know that a query to you was presented. Do you not believe that a query to you was presented?

You believe you're not imagining or hallucinating the device you're seeing my words on. You believe that arrangement of pixels isn't appearing on your screen by random chance. You believe I'm not a monkey randomly typing letters which just happen to spell out grammatically intelligible English sentences. And, most obviously, if you're saying you know a query has been presented to you, you believe a query has been presented to you.

Do you have any examples of someone knowing a statement without believing the statement?

do you have any illustrative examples?

So I simply imagine that some organism is born with their eyes naturally positioned in such a way, and so relative to the way I ordinarily see the word, they see the world upside down (and vice versa). — Michael

But isn't that our eyes? Our eyes receive light physically upside down. Our brains spin it around.

If some creature had upside down eyes relative to us, it would be up to their brain how they experience the visual orientation, not necessarily the way their eyes are positioned.

I think you're wrong to call it a sequel to the hanging paradox. It's conceptually nothing like the hanging paradox, other than the fact that both word problems both involve a hanging - which is just a superficial similarity. Conceptually it has more in common with zenos paradox of motion, which also has to do with problems of infinites and infinitely divisible units.

If A is false, it is entirely possible for B to be true; — Bob Ross

Yes, this is entirely correct

I guess what I'm saying is, the difference between depression and meditative emptiness isn't only that one is voluntary and one is forced, there's more differences than that.

Meditative emptiness is about not thinking. Depressed people think a lot. They think, I'm bored, this sucks, this isn't satisfying, I'm lonely, nothing is fulfilling, etc. The kind of emptiness that depressed people feel isn't a lack of thought - depression would be a lot more bearable for more people if it were.