Compactness Theorem- Why not Counterexample?confusion regarding compactness theoremCompactness Theorem ApplicationCompactness theorem and Tychonoff theoremLogic: compactness theorem, an exampleCompactness of Propositional LogicTwo questions about first order theories having only finite models.Why does the compactness theorem not apply to infinite subsets?Models of first-order logic and cardinalities of the domainCompactness theorem for sentential logic
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Compactness Theorem- Why not Counterexample?
confusion regarding compactness theoremCompactness Theorem ApplicationCompactness theorem and Tychonoff theoremLogic: compactness theorem, an exampleCompactness of Propositional LogicTwo questions about first order theories having only finite models.Why does the compactness theorem not apply to infinite subsets?Models of first-order logic and cardinalities of the domainCompactness theorem for sentential logic
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The compactness theorem states that if every finite subset of a set of logical statements is consistent, then the overall set of statements is consistent.
So, why is the following set of statements (each of which could be formalized under the rules of predicate logic) about a given universe not a counterexample?
- There exists at least one distinct object in our universe.
- There exist at least two distinct objects in our universe.
- There exist at least three distinct objects in our universe.
- There exist at least four distinct objects in our universe.
...
- There exist finitely many distinct objects in our universe.
Any finite subset of these statements is consistent, yet the overall set is inconsistent. Is this not a counterexample because the last sentence cannot be formalized in first-order predicate logic?
logic
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
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$begingroup$
The compactness theorem states that if every finite subset of a set of logical statements is consistent, then the overall set of statements is consistent.
So, why is the following set of statements (each of which could be formalized under the rules of predicate logic) about a given universe not a counterexample?
- There exists at least one distinct object in our universe.
- There exist at least two distinct objects in our universe.
- There exist at least three distinct objects in our universe.
- There exist at least four distinct objects in our universe.
...
- There exist finitely many distinct objects in our universe.
Any finite subset of these statements is consistent, yet the overall set is inconsistent. Is this not a counterexample because the last sentence cannot be formalized in first-order predicate logic?
logic
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
$endgroup$
3
$begingroup$
How do you formalize "there exist finitely many distinct objects in our universe"?
$endgroup$
– Wojowu
8 hours ago
3
$begingroup$
The last statement is not a first-order sentence.
$endgroup$
– Jason
8 hours ago
add a comment
|
$begingroup$
The compactness theorem states that if every finite subset of a set of logical statements is consistent, then the overall set of statements is consistent.
So, why is the following set of statements (each of which could be formalized under the rules of predicate logic) about a given universe not a counterexample?
- There exists at least one distinct object in our universe.
- There exist at least two distinct objects in our universe.
- There exist at least three distinct objects in our universe.
- There exist at least four distinct objects in our universe.
...
- There exist finitely many distinct objects in our universe.
Any finite subset of these statements is consistent, yet the overall set is inconsistent. Is this not a counterexample because the last sentence cannot be formalized in first-order predicate logic?
logic
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
$endgroup$
The compactness theorem states that if every finite subset of a set of logical statements is consistent, then the overall set of statements is consistent.
So, why is the following set of statements (each of which could be formalized under the rules of predicate logic) about a given universe not a counterexample?
- There exists at least one distinct object in our universe.
- There exist at least two distinct objects in our universe.
- There exist at least three distinct objects in our universe.
- There exist at least four distinct objects in our universe.
...
- There exist finitely many distinct objects in our universe.
Any finite subset of these statements is consistent, yet the overall set is inconsistent. Is this not a counterexample because the last sentence cannot be formalized in first-order predicate logic?
logic
logic
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
asked 8 hours ago
George BentleyGeorge Bentley
454 bronze badges
454 bronze badges
3
$begingroup$
How do you formalize "there exist finitely many distinct objects in our universe"?
$endgroup$
– Wojowu
8 hours ago
3
$begingroup$
The last statement is not a first-order sentence.
$endgroup$
– Jason
8 hours ago
add a comment
|
3
$begingroup$
How do you formalize "there exist finitely many distinct objects in our universe"?
$endgroup$
– Wojowu
8 hours ago
3
$begingroup$
The last statement is not a first-order sentence.
$endgroup$
– Jason
8 hours ago
3
3
$begingroup$
How do you formalize "there exist finitely many distinct objects in our universe"?
$endgroup$
– Wojowu
8 hours ago
$begingroup$
How do you formalize "there exist finitely many distinct objects in our universe"?
$endgroup$
– Wojowu
8 hours ago
3
3
$begingroup$
The last statement is not a first-order sentence.
$endgroup$
– Jason
8 hours ago
$begingroup$
The last statement is not a first-order sentence.
$endgroup$
– Jason
8 hours ago
add a comment
|
1 Answer
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(each of which could be formalized under the rules of predicate logic)
...
There exist finitely many distinct objects in our universe.
Are you sure that last statement is actually appropriately expressible? Indeed, the compactness theorem shows that it isn't.
(In other words, the very last line of your post is exactly right.)
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
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$begingroup$
(each of which could be formalized under the rules of predicate logic)
...
There exist finitely many distinct objects in our universe.
Are you sure that last statement is actually appropriately expressible? Indeed, the compactness theorem shows that it isn't.
(In other words, the very last line of your post is exactly right.)
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
$endgroup$
add a comment
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$begingroup$
(each of which could be formalized under the rules of predicate logic)
...
There exist finitely many distinct objects in our universe.
Are you sure that last statement is actually appropriately expressible? Indeed, the compactness theorem shows that it isn't.
(In other words, the very last line of your post is exactly right.)
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
$endgroup$
add a comment
|
$begingroup$
(each of which could be formalized under the rules of predicate logic)
...
There exist finitely many distinct objects in our universe.
Are you sure that last statement is actually appropriately expressible? Indeed, the compactness theorem shows that it isn't.
(In other words, the very last line of your post is exactly right.)
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
$endgroup$
(each of which could be formalized under the rules of predicate logic)
...
There exist finitely many distinct objects in our universe.
Are you sure that last statement is actually appropriately expressible? Indeed, the compactness theorem shows that it isn't.
(In other words, the very last line of your post is exactly right.)
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
answered 8 hours ago
Noah SchweberNoah Schweber
141k10 gold badges170 silver badges320 bronze badges
141k10 gold badges170 silver badges320 bronze badges
add a comment
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How do you formalize "there exist finitely many distinct objects in our universe"?
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– Wojowu
8 hours ago
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The last statement is not a first-order sentence.
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– Jason
8 hours ago