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Galteeth
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Taken to an extreme, do Godel's incompleteness theorems imply that the consistent mathematics we know (i.e, 2+2=4) can not encode all of reality? That certain aspects of reality do not obey conventional mathematics?
Preno said:No .
This is almost trivially false. Consider the formal system consisting of a single axiom: the statement so constructed. (And Godel's theorems has nothing to do with "reality")blkqi said:In particular, Godel constructs a true statement of reality for which no sufficiently powerful formal system can account for.
Pay attention: A formal system of a single axiom would not qualify as sufficiently powerful! The system should capture at least the notions of the Peano axioms.Hurkyl said:This is almost trivially false. Consider the formal system consisting of a single axiom: the statement so constructed.
Do I need to spell out a formal system that captures Peano's axioms and proves whatever particular statement you are considering?blkqi said:Pay attention: A formal system of a single axiom would not qualify as sufficiently powerful! The system should capture at least the notions of the Peano axioms.
No that is not clear. Such clarity would contradict Gödel's theorem which says that, roughly speaking, every "clearly true" statement of any first order formal system is provable.blkqi said:E.g. clearly Godel's statement is a true statement.
OK I misunderstood you. You can include Godel's statement in your axioms, awkward as it may be. But the Godel method can be applied to this system as well. You can add an infinite number of Godel's statements to your system but the method will never fail.Hurkyl said:Do I need to spell out a formal system that captures Peano's axioms and proves whatever particular statement you are considering?
But completeness reflects exactly a system's ability to prove as theorems every "clearly true" statement. The statement G:="G is not a theorem of system X" is true if it is not provable in system X. But it is false if it is provable in system X. A dichotomy: consistency or completeness?Hurkyl said:No that is not clear. Such clarity would contradict Gödel's theorem which says that, roughly speaking, every "clearly true" statement of any first order formal system is provable.
(the precise meaning of "clearly true" here is that it is true in every interpretation of the formal system)
It will fail if I add statements in a way that isn't recursively enumerable.blkqi said:OK I misunderstood you. You can include Godel's statement in your axioms, awkward as it may be. But the Godel method can be applied to this system as well. You can add an infinite number of Godel's statements to your system but the method will never fail.
Godel still isn't saying anything about "reality", even in terms of "common knowledge" (which, I think, you would have difficulty defining).blkqi said:Reality in this context would refer to what we accept as common knowledge. E.g. clearly Godel's statement is a true statement. You agree that I agree that you agree... etc. So we have some sort of "reality" here.
HallsofIvy said:Godel still isn't saying anything about "reality", even in terms of "common knowledge" (which, I think, you would have difficulty defining).
Godel says that in any consistent axiom system, strong enough to include the natural numbers, there must exist a statement that can neither be proved nor disproved.
Godel's Theorems, named after mathematician Kurt Godel, are two theorems in mathematical logic that have had a significant impact on the fields of mathematics and computer science. The first theorem, also known as Godel's Incompleteness Theorem, states that any formal system that is sufficiently complex cannot prove all true statements within that system. The second theorem, known as the Completeness Theorem, states that a formal system cannot prove its own consistency.
The consequences of Godel's Theorems are far-reaching and have had a profound impact on the fields of mathematics, logic, and computer science. One major consequence is that there will always be true statements about numbers that cannot be proven within a given formal system. This means that there will always be limitations to what we can prove and know using mathematics. Additionally, Godel's Theorems have raised questions about the foundations of mathematics and the nature of truth.
Godel's Theorems have challenged the idea that mathematics can be based on a set of axioms that can prove all true statements. This has led to a reevaluation of the foundations of mathematics and the development of new theories and approaches, such as category theory and homotopy type theory, that aim to address the limitations imposed by Godel's Theorems.
Godel's Theorems have had a significant impact on computer science, particularly in the field of artificial intelligence. These theorems have shown that no computer program, no matter how advanced, can ever be able to prove all true statements. This has implications for the development of artificial intelligence and has led to the exploration of alternative approaches to AI, such as neural networks and machine learning.
While Godel's Theorems have been widely accepted and have had a significant impact on multiple fields, there have been some criticisms of these theorems. Some argue that Godel's Theorems are limited in their applicability and do not necessarily apply to all formal systems. Others argue that these theorems have been misinterpreted and that they do not have the same implications for logic as they do for mathematics. However, these criticisms remain a topic of debate and do not discredit the importance of Godel's Theorems in modern mathematics and computer science.