if you could count the integers in one second, it would take infinite amount of time to count the reals. fact. qed.

@bolshoi, "p implies q" is always true if p is false, so while your statement is true, it is trivially so, and has nothing to do with your point, because you could NOT count the integers in one second, or in any amount of time (FACT). This is very awkward for you, I must say, because you have made, throughout, our ability to actually attain to a number the criterion of whether it deserves to be counted a number, but by your standards, the integers themselves should not be. (Past maybe 10^30 or so).

in fact even by your logic this statement is vacuously true, so stop denying it.

well if you're comparing size of integers vs reals you are changing the definition of size without telling anybody. so if i'm talking about counting the integers i'm changing the definition of time. there is no difference. we are both changing definitions to prove something impossible otherwise.

No, the definition of size is very precise in the case of infinite sets. Something is the same size as the integers if and only if there is a bijective function from the set to the integers.
Very well defined, see?
Your "definition of time" is just rubbish, on the other hand.

who cares about technicalities. what, do you want me to say something retarded like if you divide the amount of time to count the reals by the amount of time to count the integers you have infinity? that's what i'm trying to say, but in a more obvious way.

i understand that bijective function definition of size. but that definition is not the definition people used to use for finite sets. they came up with this definition specifically to deal with infinite sets. but you refuse to adopt a more general definition of time to deal with infinite time. you're just being a hypocrite.

No, I don't want you to say anything more about it, because you can't count either one, so the whole scenario is absurd. You should just drop the whole time-to-count argument.

Actually, bolshoi, that is the definition for finite sets. Something is of size n if and only if there is a bijection between it and the set {1,...,n}.

Even if this weren't the case, however, the definition is perfectly well-defined, so once it's in place, you can use it to see whether something is or is not countable.

"but you refuse to adopt a more general definition of time to deal with infinite time. you're just being a hypocrite. "

You haven't GIVEN me a definition, or anything like a definition. You're just saying meaningless rubbish, not defining anything.

it should be obvious that something takes more time to count if there are more elements in it. so you can say that the time it takes to count numbers in sets is equal if there exists a 1to1 bijection between the sets. hahaha.

"it should be obvious that something takes more time to count if there are more elements in it."

Oh yeah? So does it take more time to count the integers than the even integers?

"so you can say that the time it takes to count numbers in sets is equal if there exists a 1to1 bijection between the sets. hahaha. "

Problem is, you can't count either one in finite time. You're making the mistake of trying to match up cardinal numbers (set size) to real numbers (time).

if you want a more technical definition, i can't remember, but i think some mathematicians came up with turing machines that can perform a countably infinite number of operations in a finite time, then halt. so if you're talking about counting all the integers in a second, this machine could. and yes, it would still take this machine infinite time to count the reals, however, its tape wouldn't hold all the reals, so there's that issue, but even if it could, tehre would still be the problem. but i think there are models where the tape can hold reals. so you could get a technical definition, so get the pretentious stick out of your ass.

I have no idea what you're talking about anymore. Sorry.

Oh, never mind, I see.
Well, I don't really know these models you're talking about, sorry. And actually, you still haven't explained the relevance of how long it takes to count something, either, by the way. Obviously the turing machines you're talking about, if they exist at all (you've shown yourself very unreliable at interpreting mathematical literature, so I'm not going to assume they do) are purely mathematical, not mechanical constructs.

"are purely mathematical, not mechanical constructs" i would argue the same thing is true when you compare the sizes of infinite sets. what is mechanical about there being more reals than integers

Nothing. The difference is, there's something physical about time, so my point is just that you haven't demonstrated that infinitely many numbers could actually be counted off in finite time.
Actually, of course, there are proofs that they could NOT be, if you assume things like the second law of thermodynamics.

time is no more mechanical than quantity.

Wrong. Time is physical, quantity is abstract.

In any case, you still haven't given me any reason why this whole sub-argument about time EVEN MATTERS. Why are we arguing about time to count in the first place? What is your point, even if your fallacious claim were true?

i think this started because someone asked how long it would take to count the reals.

I see. OK, well, the correct answer was, "Infinite time, just like the integers."

(Of course, you couldn't count the reals even in infinite time. They're uncountable).

ok well, what i said was that even if you could count the integers in finite time, the time to count the reals would still be infinite. and you claim quantity is abstract but time isn't? both are measurable. so what exactly do you mean by abstract? for quantity you have to define what you are measuring obviously, but once that's defined, it's no more abstract than time.

Right, it's that definition in between that makes it abstract. Time, you don't have to define what you're measuring -- it's time. Quantity, could be anything. Could be time, flour, miles, anything. Hence abstract.

what about quantity of time? is that an abstract concept as well? or does specifying time render it non-abstract? if a statement is true about quantity, then it should also be true about time, since time is a particular quantity.

Well, I'm off bolshoi. Enjoy your happy adventures swimming around in your head. I suspect you will.

you know i do. i mean, the number of posts on this thread proves it.

Somebody should go through and count the number of times bolshoi changed the subject just before he got trapped.

Then slap yourself on the head because you let him get away with it.

I think at this point we all should count the number of times we responded to his posts and slap ourselves on the head that many times for being stupid enough to respond because we thought it would actually matter.