Gödel's incompleteness wrt weakend versions of ZFC

  • Thread starter phoenixthoth
  • Start date
  • Tags
    Zfc
In summary, by removing the axiom of infinity from ZFC, we would no longer satisfy the hypotheses of Gödel's incompleteness theorems. This could potentially lead to a theory that does not include arithmetical truths and cannot express elementary arithmetic. However, it is possible that this theory, referred to as ZFC-, is both consistent and complete. It is worth noting that ZFC without the axiom of infinity is equivalent to Peano Arithmetic.
  • #1
phoenixthoth
1,605
2
Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed.

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#First_incompleteness_theorem

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Second_incompleteness_theorem

We would then be in a position where the hypotheses of Gödel's theorems are not satisfied, correct? Basically, I want to remove, for the sake of argument, a minimal amount of axioms of ZFC so that ZFC minus some axiom(s) leads to a theory that does not include arithmetical truths and is not capable of expressing elementary arithmetic.

Is it possible that ZFC- (my shorthand for ZFC with some axiom(s) removed) is consistent and complete?
 
Physics news on Phys.org
  • #2
Long time no see, I am just being nostalgic...
 
  • #3
No, obviously any subtheory of an incomplete theory is also incomplete.

Also, ZFC without the axiom of infinity is basically Peano Arithmetic (or rather, replacing the Axiom of infinity by its negation yields a theory which is bi-interpretable with PA).
 

Related to Gödel's incompleteness wrt weakend versions of ZFC

1. What is Gödel's incompleteness theorem in relation to weakened versions of ZFC?

Gödel's incompleteness theorem states that any consistent formal system, such as ZFC (Zermelo-Fraenkel set theory with the axiom of choice), cannot prove all true statements within that system. This applies to weakened versions of ZFC as well, meaning that even if the system is modified or weakened, there will still be true statements that cannot be proven within that system.

2. How does Gödel's incompleteness theorem impact mathematics?

Gödel's incompleteness theorem has significant implications for mathematics because it shows that there are inherent limitations to formal systems and the ability to prove all true statements within them. This means that there will always be questions and statements that cannot be proven within a given system, leading to ongoing research and exploration in mathematics.

3. What are some examples of weakened versions of ZFC?

Some examples of weakened versions of ZFC include ZF (Zermelo-Fraenkel set theory without the axiom of choice), ZF + DC (Zermelo-Fraenkel set theory with the axiom of dependent choice), and ZF + AD (Zermelo-Fraenkel set theory with the axiom of determinacy). These versions of ZFC have been modified or weakened in different ways, but they still fall under the scope of Gödel's incompleteness theorem.

4. Can Gödel's incompleteness theorem be proven within ZFC?

No, Gödel's incompleteness theorem cannot be proven within ZFC or any other formal system. This is because the theorem itself states that there will always be true statements that cannot be proven within a given system, and attempting to prove the theorem within that system would contradict it.

5. How does Gödel's incompleteness theorem relate to the foundations of mathematics?

Gödel's incompleteness theorem has significant implications for the foundations of mathematics, as it shows that there are inherent limitations to formal systems and the ability to prove all true statements within them. This challenges the idea of a complete and consistent foundation for mathematics, and has led to ongoing research and debate within the field.

Similar threads

Incompleteness of Euclidean Geometry: Proving the Parallel Postulate
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
A confusion about Godel theorem and real numbers
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
2K
Doubting the effects of Gödel's incompleteness theorems on Physics
    • Other Physics Topics
Replies
2
Views
2K
Decoding Gödel Numbers for Sequences: A Constructive Approach
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
Is Alan Turing given too much credit when it comes to computers?
    • Programming and Computer Science
Replies
29
Views
3K
Q.E.D.Undecidability of Q: Proving Godel's First Incompleteness Theorem
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
953
Mathematica This Week's Finds in Mathematical Physics (Week 226)
    • MATLAB, Maple, Mathematica, LaTeX
Replies
7
Views
2K
Mathematica This Week's Finds in Mathematical Physics (Week 226)
    • MATLAB, Maple, Mathematica, LaTeX
Replies
7
Views
3K
Mathematica This Week's Finds in Mathematical Physics (Week 244)
    • MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
2K
Mathematica This Week's Finds in Mathematical Physics (Week 230)
    • MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top