As the discussion on my previous blog post got rolling, Vinaire posted an excellent comment that I thought warranted a blog post on its own.
- Kurt Gödel
- Gödel’s incompleteness theorem simply explained
- Gödel’s ontological proof – a must read
- A proof against determinism?
Godel’s incompleteness theorem applies only to axiomatic systems capable of doing arithmetic. I do not know if Godel’s argument can be extended to as complex a system as the universe.
1. the doctrine that all facts and events exemplify natural laws.
2. the doctrine that all events, including human choices and decisions, have sufficient causes.
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.
A set of axioms is complete if, for any statement in the axioms’ language, either that statement or its negation is provable from the axioms.
A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms.
e·nu·mer·ate verb (used with object)
1. to mention separately as if in counting; name one by one; specify, as in a list: Let me enumerate the many flaws in your hypothesis.
2. to ascertain the number of; count.
A formal theory is said to be effectively generated if there is a computer program that, in principle, could enumerate all the axioms of the theory without listing any statements that are not axioms. This is equivalent to the existence of a program that enumerates all the theorems of the theory without enumerating any statements that are not theorems.
Gödel’s first incompleteness theorem states that:
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory…
Gödel’s theorem shows that, in theories that include a small portion of number theory, a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent.
There are complete and consistent lists of axioms for arithmetic that cannot be enumerated by a computer program. For example, one might take all true statements about the natural numbers to be axioms (and no false statements), which gives the theory known as “true arithmetic”. The difficulty is that there is no mechanical way to decide, given a statement about the natural numbers, whether it is an axiom of this theory, and thus there is no effective way to verify a formal proof in this theory.
This may mean that if this universe (with both its physical and spiritual aspects) can be expressed through a consistent set of principles, then there is a truth about this universe that cannot be demonstrated using those set of principles. That truth may look at this universe (as a whole) exactly for what it is. Such a truth may not be derivable from the set of principles that supposedly describe the universe.
Gödel’s second incompleteness theorem states that:
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.
The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that could be formalized in the theory that is proved consistent. The second incompleteness theorem is similar to the Liar’s paradox, “This sentence is false,” which contains an inherent contradiction about its truth value.
This may mean that this universe cannot contain the ultimate truth about itself. The ultimate truth is unknowable from the reference point of this universe.
If we go by the definition of determinism that all facts and events exemplify natural laws, we cannot say for certain if that is true or not. In other words, not everything may be predictable ahead of its occurrence.
Manifestations may be related to each other in strict logical sequence meaning that any manifestation may be shown to follow from another manifestation. However, it may be impossible to determine how a manifestation may come to be on its own. This is another version of saying, “Absolutes are unattainable.”
So a system may be deterministic only in a relative sense. It can neither be absolutely deterministic, nor can it be absolutely non-deterministic.
Link to article on Vinaire’s blog: Gödel and Determinism