Consequences of Godel's Theorems

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Discussion Overview

The discussion revolves around the implications of Gödel's incompleteness theorems, particularly whether these theorems suggest that conventional mathematics can fully encode reality. Participants explore the relationship between formal systems, truth, and provability, raising questions about the nature of mathematical statements and their connection to reality.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that Gödel's theorems imply that consistent mathematics cannot encode all aspects of reality, suggesting that certain truths may exist outside formal systems.
  • Others argue that Gödel's theorems do not pertain to "reality" and emphasize that the theorems are about decidability within formal systems.
  • A participant mentions that Gödel constructs a true statement that cannot be accounted for by sufficiently powerful formal systems, while another counters that this is trivially false under certain conditions.
  • There is a discussion about the clarity of Gödel's statement and its implications for provability, with some asserting that clarity contradicts Gödel's theorem.
  • Some participants challenge the notion of including Gödel's statement in axioms, arguing about the implications of adding such statements to formal systems.
  • One participant highlights that Gödel's theorems indicate the existence of statements that cannot be proven or disproven within a consistent system strong enough to include natural numbers.
  • There is a debate over the definition of "reality" in the context of Gödel's statements, with some suggesting it refers to common knowledge.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the implications of Gödel's theorems and their relationship to reality. The discussion remains unresolved, with no consensus on the interpretations of Gödel's work or its implications.

Contextual Notes

Participants note limitations in defining "reality" and the conditions under which Gödel's statements can be considered true or false. There is also mention of the need for a sufficiently powerful formal system to engage with Gödel's theorems, which remains a point of contention.

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?
 
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No .
 
Preno said:
No .

Care to elaborate?
 
In particular, Godel constructs a true statement of reality for which no sufficiently powerful formal system can account for. It's construction is simply for the reason of stumping formal systems.

But you should know something else: I am a liar.
 
blkqi said:
In particular, Godel constructs a true statement of reality for which no sufficiently powerful formal system can account for.
This is almost trivially false. Consider the formal system consisting of a single axiom: the statement so constructed. :-p (And Godel's theorems has nothing to do with "reality")
 
Hurkyl said:
This is almost trivially false. Consider the formal system consisting of a single axiom: the statement so constructed. :-p
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.
 
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.
 
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.
Do I need to spell out a formal system that captures Peano's axioms and proves whatever particular statement you are considering?
 
blkqi said:
E.g. clearly Godel's statement is a true statement.
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)
 
  • #10
Hurkyl said:
Do I need to spell out a formal system that captures Peano's axioms and proves whatever particular statement you are considering?
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.
 
  • #11
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)
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?
 
  • #12
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.
It will fail if I add statements in a way that isn't recursively enumerable.

But that's not the point. That Gödel constructs an "unprovable truth" is a myth -- a misinterpretation of what the theorem states. Or, possibly, a biased interpretation of what the theorem states.

For a (sufficiently strong, but not too strong) formal theory, Gödel constructs a statement that can not be proven within that theory.

Whether that statement is true or false, something in-between, or even whether it's meaningful to ask that question is a matter of semantics -- of interpretation.

One thing's for certain: there exists a set-theoretic model of the theory in which the statement is false. (also there also exists a set-theoretic model of the theory in which the statement is true)
 
  • #13
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.
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.
 
  • #14
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.

This is what I am asking. At it's bare bones, isn't conventional math an axiomatic system?
 
  • #15
godel's theorems are about decidability in certain formal systems.
they have nothing to do with "encoding reality"... whatever that means.
 

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