Infinity is an abstraction that makes thinking about certain problems easier. It’s not a feature of reality (=territory), it’s a tool for thinking (=map). As with any tool, there are contexts where it is very helpful; there are contexts where it is inappropriate; and there are contexts where it can be misused.
Without the concept of infinity you’ll struggle with even basic geometry (consider a line) and calculus becomes an outright impossibility.
Without the concept of infinity you’ll struggle with even basic geometry
No matter, no matter. The fact that this concept of infinity sometimes makes our life easier, doesn’t make everything right about it. Maybe we should try harder. Going the right path isn’t always easy, it can be extremely hard. Perhaps unbelievably hard.
100 years ago, they thought that the self reflection is in the core of paradoxes, like Russell’s and related.
I thought long ago, that the infinity has its share here. With Yablo’s paradox, this is quite obvious. No self-reflection, but still a paradox. But if the Yablo’s list of statements is finite, than the last statement is true (vacuous truth), and—no problemo!
I can’t live with an infinite number of (even all finite) sentences. Because those statements may all be “every higher on this list statement is false”. But that would imply that the next statement is in fact true. Which means that this statement is false. But all saying the same, that would mean it is true. And so on in a circle.
I may have misunderstood your argument Thomas. Are you saying that because it’s possible to construct a paradox ( in this case Yablo’s paradox ) using an infinitude, that the concept of infinity is itself paradoxical?
Couldn’t you make a similar argument about finite systems such as, say?:
A: B is false,
B: A is false
Here are only two sentences. Is the number two therefore paradoxical? I apologize if it sounds like I’m trying to parody your argument—I really would like to learn that I’ve misunderstood it, and in what way.
using an infinitude, that the concept of infinity is itself paradoxical?
I have to admit, that no. Perhaps it is something else wrong here. It might be. Not very likely, but still possible. We see, that finite lists don’t suffer because of Yablo’s paradox, infinite lists do. Still, it may be something else which really causes the A & NOT A situation..
Couldn’t you make a similar argument about finite systems such as, say?:
A: B is false, B: A is false
You can. And you have a contradictory little system here. Which is therefore useless. So you take another one, like the rules for the game of chess. Hoping, that there is no paradox there. (But there is. You can’t believe it, but there is a paradox inside the chess rules!)
In practice, you can continue to play chess and use infinities. But that is at your own risk.
No. Zeno paradoxes aren’t real. Why? You have to have an axiomatic system and inside this system you must be able to prove A & NOT A. Then and only then, there is a paradox.
Then EVERY statement is provable inside this axiomatic system and the system is useless.
Zeno had only “paradoxes”. Had he formulated one of his paradoxes inside geometry, that would be something! But he didn’t. All Zeno had was his intuition that “you can’t do infinite number of steps in a finite time”.
For the Russell’s paradox inside the Naive Set Theory, that is a different story. Using those axioms one can prove that his B is its own member. And one can also prove that it isn’t.
But those Cantor’s examples, like the Infinite Hotel—are NOT paradoxes. It is odd, and “paradoxical” if you wish, but that’s fine. Had Cantor proved, that this hotel can accommodate exactly 7 guests - and someone else (or Cantor) had proved that this hotel can accommodate exactly 12 guests—that would be a paradox indeed.
“you can’t do infinite number of steps in a finite time”
Well, can you? If some finite period must elapse when a finite distance is covered, an an infinite distance is greater than any finite distance, then the period of time elapsed in crossing an infinite segment must be greater than the period that elapses for crossing any finite segment, and thus also infinite.
I suppose you can also assume that you can cross a finite segment without a finite period of time elapsing—but then what’s to prevent any finite segment of arbitrary length being crossed instantaneously?
What seemed the infinite number of steps to Zeno (and pretty much to everybody else), may be only some finite number of Planck’s lengths to cross in some finite number of Planck’s time units.
(In the fifth century A.D., Indian mathematicians reconciled the doubts of Zeno, using infinite series and those solutions officially still hold. If you want to keep the infinitely divisible space and time, you may do it their way.)
He proved, that every system rich enough to contain “infinite arithmetics” is EITHER inconsistent (have paradoxes) EITHER have some non provable sentences.
Almost everybody thought—“Okay, okay, so we’ll always have nonprovables. It’s a shame, but what can we do?”
But this was not the only one explanation. The “Okay, okay so we can’t make the complete “infinite arithmetic” without a paradox. We must cease to even try that.”—flies at least as well.
He proved, that there is ALWAYS either at least one “A & ~A” (and therefore many) - either an unprovable theorem exists. Inside all those systems, which contain the “standard calculus”!
He didn’t prove an actual “A & ~A”, but that one always exists, if there are no unprovable theorems in those “standard calculus systems”.
He proved, that there is ALWAYS either at least one “A & ~A” (and therefore many) - either an unprovable theorem exists. Inside all those systems, which contain the “standard calculus”!
That isn’t actually grammatical English, and unfortunately some plausible guesses at reconstructing it produce things that are completely false. So here’s what Goedel actually he did: he proved that for any powerful enough system either there is at least one (hence many) A & ~A, or there are A for which neither A nor ~A can be proved by the system.
It’s not “almost correct”, it’s actually correct. I wasn’t trying to make it complete :-). Yes, the “powerful enough” criterion is about being able to represent arithmetic. But if you want an actually-complete statement then “with arithmetics” and “contain the ‘standard calculus’” really aren’t any better than “powerful enough”. (I can never remember the exact requirements and am too lazy to look them up for this sort of discussion, and it’s not like the details matter very much here.)
Finite sets of numbers have obvious problems, well explored in computer science (if you integer has be be expressed in 4 bytes, you can express only a finite set of integers. Or real numbers, for that matter).
Trivially, if your finite set ends at n, what is the sum of (n-1) + (n-2)? A plausible answer is “you can’t do that”, but that’s also an answer to any paradox whatsoever.
How do you feel about imaginary numbers, by the way?
A plausible answer is “you can’t do that”, but that’s also an answer to any paradox whatsoever.
No, this is not true. The only answer we have to paradoxes is—change you axiom system!
How do you feel about imaginary numbers, by the way?
There are no paradoxes there. At least no paradoxes which wouldn’t be present in the real numbers already.
SQRT(-1)=i is no paradox whatsoever. People might often say—“oh, well that’s paradoxical”.
No, it’s not. No A & ~ A here. It might be weird to somebody, that you can calculate square roots from negative numbers. I really don’t see why, but it might.
Weirdness and paradoxicality are two different things.
You can eliminate paradoxes by declaring operations which lead to them “illegal”
Precisely! You narrow your axiomatic system, hoping that one of contradictors will fall out. They’ve eliminated self containing sets from the Naive Set Theory and Russell’s paradox vanished. Hopefully.
Is i a number? If not, what is it?
It is not “what is it”, it is “how does it behave”. Whatever behaves as a number, is a number.
Does it have a sign?
If it behaves as it had a sign … And it does. So yes, it has a sign!
Infinity is an abstraction that makes thinking about certain problems easier. It’s not a feature of reality (=territory), it’s a tool for thinking (=map). As with any tool, there are contexts where it is very helpful; there are contexts where it is inappropriate; and there are contexts where it can be misused.
Without the concept of infinity you’ll struggle with even basic geometry (consider a line) and calculus becomes an outright impossibility.
No matter, no matter. The fact that this concept of infinity sometimes makes our life easier, doesn’t make everything right about it. Maybe we should try harder. Going the right path isn’t always easy, it can be extremely hard. Perhaps unbelievably hard.
100 years ago, they thought that the self reflection is in the core of paradoxes, like Russell’s and related.
I thought long ago, that the infinity has its share here. With Yablo’s paradox, this is quite obvious. No self-reflection, but still a paradox. But if the Yablo’s list of statements is finite, than the last statement is true (vacuous truth), and—no problemo!
I can’t live with an infinite number of (even all finite) sentences. Because those statements may all be “every higher on this list statement is false”. But that would imply that the next statement is in fact true. Which means that this statement is false. But all saying the same, that would mean it is true. And so on in a circle.
I see a real paradox here, not a curiosum.
I may have misunderstood your argument Thomas. Are you saying that because it’s possible to construct a paradox ( in this case Yablo’s paradox ) using an infinitude, that the concept of infinity is itself paradoxical?
Couldn’t you make a similar argument about finite systems such as, say?:
A: B is false, B: A is false
Here are only two sentences. Is the number two therefore paradoxical? I apologize if it sounds like I’m trying to parody your argument—I really would like to learn that I’ve misunderstood it, and in what way.
I have to admit, that no. Perhaps it is something else wrong here. It might be. Not very likely, but still possible. We see, that finite lists don’t suffer because of Yablo’s paradox, infinite lists do. Still, it may be something else which really causes the A & NOT A situation..
You can. And you have a contradictory little system here. Which is therefore useless. So you take another one, like the rules for the game of chess. Hoping, that there is no paradox there. (But there is. You can’t believe it, but there is a paradox inside the chess rules!)
In practice, you can continue to play chess and use infinities. But that is at your own risk.
On which basis do you decide which path is right and which is not?
Are Zeno’s paradoxes also “real”?
No. Zeno paradoxes aren’t real. Why? You have to have an axiomatic system and inside this system you must be able to prove A & NOT A. Then and only then, there is a paradox.
Then EVERY statement is provable inside this axiomatic system and the system is useless.
Zeno had only “paradoxes”. Had he formulated one of his paradoxes inside geometry, that would be something! But he didn’t. All Zeno had was his intuition that “you can’t do infinite number of steps in a finite time”.
For the Russell’s paradox inside the Naive Set Theory, that is a different story. Using those axioms one can prove that his B is its own member. And one can also prove that it isn’t.
But those Cantor’s examples, like the Infinite Hotel—are NOT paradoxes. It is odd, and “paradoxical” if you wish, but that’s fine. Had Cantor proved, that this hotel can accommodate exactly 7 guests - and someone else (or Cantor) had proved that this hotel can accommodate exactly 12 guests—that would be a paradox indeed.
Assuming that both proofs were correct.
I know, you know that.
“you can’t do infinite number of steps in a finite time”
Well, can you? If some finite period must elapse when a finite distance is covered, an an infinite distance is greater than any finite distance, then the period of time elapsed in crossing an infinite segment must be greater than the period that elapses for crossing any finite segment, and thus also infinite.
I suppose you can also assume that you can cross a finite segment without a finite period of time elapsing—but then what’s to prevent any finite segment of arbitrary length being crossed instantaneously?
What seemed the infinite number of steps to Zeno (and pretty much to everybody else), may be only some finite number of Planck’s lengths to cross in some finite number of Planck’s time units.
(In the fifth century A.D., Indian mathematicians reconciled the doubts of Zeno, using infinite series and those solutions officially still hold. If you want to keep the infinitely divisible space and time, you may do it their way.)
So, Gödel?
He proved, that every system rich enough to contain “infinite arithmetics” is EITHER inconsistent (have paradoxes) EITHER have some non provable sentences.
Almost everybody thought—“Okay, okay, so we’ll always have nonprovables. It’s a shame, but what can we do?”
But this was not the only one explanation. The “Okay, okay so we can’t make the complete “infinite arithmetic” without a paradox. We must cease to even try that.”—flies at least as well.
I don’t know about that. They key word is “useful”. I’m not quite ready to discard and forget Peano arithmetic.
Yes, it is useful. Like a buggy program may be useful. But there comes the time of refactoring the whole thing.
Perhaps on an entirely new architecture.
What “A & ~A” did he prove?
He proved, that there is ALWAYS either at least one “A & ~A” (and therefore many) - either an unprovable theorem exists. Inside all those systems, which contain the “standard calculus”!
He didn’t prove an actual “A & ~A”, but that one always exists, if there are no unprovable theorems in those “standard calculus systems”.
That isn’t actually grammatical English, and unfortunately some plausible guesses at reconstructing it produce things that are completely false. So here’s what Goedel actually he did: he proved that for any powerful enough system either there is at least one (hence many) A & ~A, or there are A for which neither A nor ~A can be proved by the system.
That is almost correct. But “powerful enough” isn’t clear at all at your statement.
“Powerful enough” means those infinite sets with arithmetics.
It’s not “almost correct”, it’s actually correct. I wasn’t trying to make it complete :-). Yes, the “powerful enough” criterion is about being able to represent arithmetic. But if you want an actually-complete statement then “with arithmetics” and “contain the ‘standard calculus’” really aren’t any better than “powerful enough”. (I can never remember the exact requirements and am too lazy to look them up for this sort of discussion, and it’s not like the details matter very much here.)
Not only arithmetic, but an arithmetic with arbitrary large numbers is required for the Godel’s Incompleteness Theorem.
You need at least “all integers”, that is an infinite set.
Maybe some incompleteness can be proved for the finite sets as well. But it’s not known to be so. Nobody proved that yet.
Finite sets of numbers have obvious problems, well explored in computer science (if you integer has be be expressed in 4 bytes, you can express only a finite set of integers. Or real numbers, for that matter).
Trivially, if your finite set ends at n, what is the sum of (n-1) + (n-2)? A plausible answer is “you can’t do that”, but that’s also an answer to any paradox whatsoever.
How do you feel about imaginary numbers, by the way?
No, this is not true. The only answer we have to paradoxes is—change you axiom system!
There are no paradoxes there. At least no paradoxes which wouldn’t be present in the real numbers already.
SQRT(-1)=i is no paradox whatsoever. People might often say—“oh, well that’s paradoxical”.
No, it’s not. No A & ~ A here. It might be weird to somebody, that you can calculate square roots from negative numbers. I really don’t see why, but it might.
Weirdness and paradoxicality are two different things.
I don’t mind weirdness, I hate paradoxicality.
You can eliminate paradoxes by declaring operations which lead to them “illegal” (or their outcomes to be undefined).
Is i a number? If not, what is it? Does it have a sign?
Precisely! You narrow your axiomatic system, hoping that one of contradictors will fall out. They’ve eliminated self containing sets from the Naive Set Theory and Russell’s paradox vanished. Hopefully.
It is not “what is it”, it is “how does it behave”. Whatever behaves as a number, is a number.
If it behaves as it had a sign … And it does. So yes, it has a sign!
Which is what?
Everything which behaves like a sign.