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permuted and rephrased based on comment thread.
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Suvrit
Suvrit
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I was happily surfing the arXiv, when I was jolted byUpdate (21st April, 2019). Removed the following paper:

Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity byreference M. Kim, Mar. 2012/ initial trigger behind my question (please see comment thread below for the reasons).

Abstract. This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

Now this seems to be a very major claim Am retaining, and I lack the background to be able to judge ifof course, the claim is trueactual question, or there is some subtle or even obvious defectnoted both in the paper's argumentstitle as well as in the post below. But picking on

The key questions of this paper itself is notpost are the purpose of my question.following:

1. The paper, however, got me very curious about howHow "disastrous" would inconsistency of ZFC really be?

2. A slightly more preciserefined question is: what would be the major consequences of the kind of inconsistency claimed in the abstract cited abovewhat would be the major consequences of different types of alleged inconsistencies in ZFC?

 

If you feel that my questions might not admit "clearly right" answers, I will be happy to make this post CW.Old material (the "no-longer relevant" part of the question).

I was happily surfing the arXiv, when I was jolted by the following paper:

Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by M. Kim, Mar. 2012.

Abstract. This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

Now this seems to be a very major claim, and I lack the background to be able to judge if the claim is true, or there is some subtle or even obvious defect in the paper's arguments. But picking on this paper itself is not the purpose of my question.

If you feel that my questions might not admit "clearly right" answers, I will be happy to make this post CW.

I was happily surfing the arXiv, when I was jolted by the following paper:

Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by M. Kim, Mar. 2012.

Abstract. This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

Now this seems to be a very major claim, and I lack the background to be able to judge if the claim is true, or there is some subtle or even obvious defect in the paper's arguments. But picking on this paper itself is not the purpose of my question.

1. The paper, however, got me very curious about how "disastrous" would inconsistency of ZFC really be?

2. A slightly more precise question is: what would be the major consequences of the kind of inconsistency claimed in the abstract cited above?

If you feel that my questions might not admit "clearly right" answers, I will be happy to make this post CW.

Update (21st April, 2019). Removed the reference / initial trigger behind my question (please see comment thread below for the reasons). Am retaining, of course, the actual question, noted both in the title as well as in the post below.

The key questions of this post are the following:

1. How "disastrous" would inconsistency of ZFC really be?

2. A slightly more refined question is: what would be the major consequences of different types of alleged inconsistencies in ZFC?

 

Old material (the "no-longer relevant" part of the question).

I was happily surfing the arXiv, when I was jolted by the following paper:

Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by M. Kim, Mar. 2012.

Abstract. This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

Now this seems to be a very major claim, and I lack the background to be able to judge if the claim is true, or there is some subtle or even obvious defect in the paper's arguments. But picking on this paper itself is not the purpose of my question.

If you feel that my questions might not admit "clearly right" answers, I will be happy to make this post CW.

Source Link
Suvrit
Suvrit
  • 28.6k
  • 7
  • 82
  • 150

What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper:

Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by M. Kim, Mar. 2012.

Abstract. This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

Now this seems to be a very major claim, and I lack the background to be able to judge if the claim is true, or there is some subtle or even obvious defect in the paper's arguments. But picking on this paper itself is not the purpose of my question.

1. The paper, however, got me very curious about how "disastrous" would inconsistency of ZFC really be?

2. A slightly more precise question is: what would be the major consequences of the kind of inconsistency claimed in the abstract cited above?

If you feel that my questions might not admit "clearly right" answers, I will be happy to make this post CW.