Infinity, Gödels theorem, determinism and self reference Watch

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Jonatan
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As of you may know, Gödels theorem states that no formal axiomatic system capable of represanting all the natural integers can be both consistent and complete, however a recent discussion regarding the concept of infinity made me wounder if this theorem has any relevance in practice.

Considder a finite space of the universe. It contains a finite number of particles, all of which has a finite energy. Thus there is a limit to the maximum entropy such a system may posess. As the entropy of the system is limited to a finite degree, it can only contain a finite number of micro states. And thus only a finite amount o information can be contained within it.

Thus no finite amount of space can contain an infinite amount of information. This limits the axiomatic formal systems we may create inside such a system to only contain a finite number of theorems (Since when we run out of places to store information, we will be forced to delete an old theorem in order to store the new one). Therefore, no man made formal system can ever produce enough theorems for Gödels theorem to be relevant. Therefore, it seems, a finite system could (in principle ) be deterministic.

However, the limitation of information storage hit us once again. Since our brains are certainly a part of the system which we try to predict the behaviour of (remember that Heisenbergs uncertainty principle prevents us from observing a system without at the same time disturbing it) the system will not have enough possible microstates to store all the information about the system and at the same time perfomr the calculation predicting the behaviour of the system storing it in a format we may undertsant. Thus predicting the future with infalable accuracy seems to be theoretically impossible.
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bananaman
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(Original post by Jonatan)
As of you may know, Gödels theorem states that no formal axiomatic system capable of represanting all the natural integers can be both consistent and complete, however a recent discussion regarding the concept of infinity made me wounder if this theorem has any relevance in practice.

Considder a finite space of the universe. It contains a finite number of particles, all of which has a finite energy. Thus there is a limit to the maximum entropy such a system may posess. As the entropy of the system is limited to a finite degree, it can only contain a finite number of micro states. And thus only a finite amount o information can be contained within it.

Thus no finite amount of space can contain an infinite amount of information. This limits the axiomatic formal systems we may create inside such a system to only contain a finite number of theorems (Since when we run out of places to store information, we will be forced to delete an old theorem in order to store the new one). Therefore, no man made formal system can ever produce enough theorems for Gödels theorem to be relevant. Therefore, it seems, a finite system could (in principle ) be deterministic.

However, the limitation of information storage hit us once again. Since our brains are certainly a part of the system which we try to predict the behaviour of (remember that Heisenbergs uncertainty principle prevents us from observing a system without at the same time disturbing it) the system will not have enough possible microstates to store all the information about the system and at the same time perfomr the calculation predicting the behaviour of the system storing it in a format we may undertsant. Thus predicting the future with infalable accuracy seems to be theoretically impossible.
No you didn't, you just read a fairly recent article of Stephen Hawking's. Anyway, determinism in physics is largely undermined by the fact that the quantum mechanical world of very small things appears to work randomly and probablistically, rather than as cogs in a big machine.
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GeneralGrievous
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where is my reply to this? why was it deleted?



ah, sorry, didnt see that jonothan had posted 2 topics on the same thing :rolleyes:
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Jonatan
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(Original post by bananaman)
No you didn't, you just read a fairly recent article of Stephen Hawking's. Anyway, determinism in physics is largely undermined by the fact that the quantum mechanical world of very small things appears to work randomly and probablistically, rather than as cogs in a big machine.
I have not read that article by hawkings. Could you give me a link as I find it interesting.

Also quantum mechanics make the position and momentum of a particle indeterminable, but the probabilities are still determinable. If you considder the electrons to be waves rather than particles with a defenite position and momentum you can actually predict the wavefunctions (This Hawking did mention in universe in a nutshell) so you may say that it is deterministic anyways. However, my point was that since the number of possibel microstates in a system is limited so are teh number of theorems producible and thus Gödels theorem does not apply.
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Howard
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(Original post by Jonatan)
As of you may know, Gödels theorem states that no formal axiomatic system capable of represanting all the natural integers can be both consistent and complete, however a recent discussion regarding the concept of infinity made me wounder if this theorem has any relevance in practice.

Considder a finite space of the universe. It contains a finite number of particles, all of which has a finite energy. Thus there is a limit to the maximum entropy such a system may posess. As the entropy of the system is limited to a finite degree, it can only contain a finite number of micro states. And thus only a finite amount o information can be contained within it.

Thus no finite amount of space can contain an infinite amount of information. This limits the axiomatic formal systems we may create inside such a system to only contain a finite number of theorems (Since when we run out of places to store information, we will be forced to delete an old theorem in order to store the new one). Therefore, no man made formal system can ever produce enough theorems for Gödels theorem to be relevant. Therefore, it seems, a finite system could (in principle ) be deterministic.

However, the limitation of information storage hit us once again. Since our brains are certainly a part of the system which we try to predict the behaviour of (remember that Heisenbergs uncertainty principle prevents us from observing a system without at the same time disturbing it) the system will not have enough possible microstates to store all the information about the system and at the same time perfomr the calculation predicting the behaviour of the system storing it in a format we may undertsant. Thus predicting the future with infalable accuracy seems to be theoretically impossible.
Well, you've lost me completely.
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pedy1986
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Could you not have put this in laymans terms?
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Howard
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(Original post by corey)
Could you not have put this in laymans terms?
Yes, not very digestable is it?
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