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Member (Idle past 2324 days) Posts: 2870 From: Limburg, The Netherlands Joined: |
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Author | Topic: 0.99999~ = 1 ? | |||||||||||||||||||||||
Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
Yeah, but finitism is silly.
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cavediver Member (Idle past 3672 days) Posts: 4129 From: UK Joined: |
Yeah, but finitism is silly. I'll not disagree Edited by cavediver, : No reason given.
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cavediver Member (Idle past 3672 days) Posts: 4129 From: UK Joined: |
Ah yes. Even if every digit were the size of only 1 planck volume, the observable universe would still not be big enough to contain a digital representation of it It's far worse than that! Let's say that for each Planck volume, you are given another entire universe!! And for each Planck volume in that universe, you are given *another* universe!! You would still not have enough Planck volumes to even begin to write out Graham's number
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Dr Adequate Member (Idle past 313 days) Posts: 16113 Joined: |
That is, two numbers a and b are equal if there is no number, e such that |a - b| < e Read through that again. You meant to say "if for every (positive) number e, |a - b| < e". Or possibly you meant to say "if there is no (positive) number e such that |a - b| > e". Edited by Dr Adequate, : No reason given.
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Dr Adequate Member (Idle past 313 days) Posts: 16113 Joined: |
Surely the difference between 0.999R and 1 is practically non-existant and philosophically massive? It is the difference between claiming complete certainty (for example) and always allowing for the possibility of that which is unexpected (no matter how likely or unlikely). It is the difference between an obtainable destination and that which can never exist or be obtained in reality. I am no mathematician. But surely the difference between 1 and NOT 1 is as significant as ever. No matter what the NOT 1 may be? It's funny, no-one has any difficulty with grasping that 1/2 and 2/4 are the same thing. Well, here we have a similar situation. We happen to use a system of notation in which sometimes there is more than one way to write exactly the same number. Edited by Dr Adequate, : No reason given.
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Son Goku Inactive Member |
Ah, Graham's number isn't that big, it's less than Graham's number + 1. In fact most numbers are probably bigger than it.
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Son Goku Inactive Member |
This whole Constructivist thing confuses me, I hope you can help.
Surely constructivism assumes a somewhat Platonist view. If you are being a total formalist then the rules/axioms of infinite series are consistent and that's all it means for 0.999.... to exist. That is, it is a game of symbols and rules which doesn't contradict itself. In order to say 0.9999.... doesn't exist, you need to think there was an objective mathematical reality and only constructive methods get you there. If you know, what do constructivists think of standard math? Do they believe what is constructive is Platonically true and anything else is just formalist games with pen and paper? Usually I would be skeptical about such a philosophy, however Edward Nelson is a constructivist and he's no fool. (He took the first real hard steps toward showing Quantum Field Theory makes sense mathematically.)
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Well It appears that the genaral consensus amongst those who are more mathematically literate than myself is that I am wrong.
In which case I stand corrected. And in which case I will need to find a new nomenclature for expressing all but certain without the philosophical possibility of complete and absolute certainty. But just to be clear is it false to say that 0.999R < 1?
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
This is a genuine question. As it appears that my understanding up until now has been incomplete (to put it politely - or just wrong if we are to be more blunt.
But is it wrong to say that 0.999R <1?
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Nope, they're completely the same, philosophically and otherwise. Well up until now I have (rightly or wrongly) been expressing the difference between probablistic certainty or otherwise as the differnce between 1 and NOT 1. Where (in my mind at least) 0.999R is not equal to 1. It appears I am need to change my nomenclature. But I am not sure what I should use instead?
Nope. There is no uncertainty, 0.9999~ is 1. In fact all decimal representations are infinite sequences, it's just some of them finish in an infinite number of 0s. This is not, as it may sound, a cute aphorism but actually fundamental to the construction of the real numbers. Real numbers are limits of inifnite sequences. Well I get what you are saying. I think. But it still seems counter-intuitive that 1/0.999R is entirely equal to 1/1. For example.
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PaulK Member Posts: 17828 Joined: Member Rating: 2.3 |
quote: I'm afraid that it is. Unless you can show that 1 - 0.999R is greater than zero. (Hint: What is the decimal expansion of 1 - 0.999R. How far do you have to go to find a non-zero digit ?)
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
Double post. See below.
Edited by Straggler, : No reason given.
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Straggler Member (Idle past 95 days) Posts: 10333 From: London England Joined: |
I'm afraid that it is. Yeah I am starting to get that despite inbuilt resistance.
Unless you can show that 1 - 0.999R is greater than zero. Can it be shown mathematically that 1 - 0.999R is equal to zero?
(Hint: What is the decimal expansion of 1 - 0.999R. How far do you have to go to find a non-zero digit ?) One step further down the infinite chain than you need to go to make 0.999R the same as 1?
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Huntard Member (Idle past 2324 days) Posts: 2870 From: Limburg, The Netherlands Joined: |
Straggler writes:
Yes, since they are the same.
Can it be shown mathematically that 1 - 0.999R is equal to zero? One step further down the infinite chain than you need to go to make 0.999R the same as 1?
How many steps does an infinte chain contain then? And what is infinity + 1? I hunt for the truth I am the one Orgasmatron, the outstretched grasping handMy image is of agony, my servants rape the land Obsequious and arrogant, clandestine and vain Two thousand years of misery, of torture in my name Hypocrisy made paramount, paranoia the law My name is called religion, sadistic, sacred whore. -Lyrics by Lemmy Kilmister of Motorhead
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PaulK Member Posts: 17828 Joined: Member Rating: 2.3 |
quote: Yes we can show that there is no number between it and zero, for instance (you can get that from the fact that there is no number between 1 and 0.999R)
quote: Nice attempt to turn it around. However, would you agree that 1 - 0.999R is zero to an infinite precision ?
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