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What is the point of Löwenheim–Skolem(Compactness theorem in the) Overspill principle?

The principle (Theorem statescalled a Löwenheim–Skolem theorem by Huth and Ryan  ) states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Edit As sjmc remarks, the statement of the theorem is actually a consecuence of it, known as Overspill Principle. The "Logic in Computer Science" misinforms us!

Nevertheless, I started to understand why people (mistakenly) ask me to differentiate between infinite amount of models and models of infinite size. They fail to recognize that principle "The fact that I can always name a number larger than yours implies that we have an infinite model/set", which is intuitive and used to prove the overspill theorem, also implies that the model of infinite size exists. The set of models $A = \{M_k, M_l, M_m \ldots\}$ ofhas sizes $S = \{k, l, m, \ldots\}$ korrespondinglycorrespondingly. When you speak about size of models, you basically speak of the numbers in $S$. When you say "a model of size n" you just say "n". Thus, we can forget about set of models and speak only about S. Now, you say that "whatever integer you have, set S contains a larger one." This basically means that S contains an infinite number (i.e. $A$ contains an enumerable modelinfinite models). What to be proven here?

In other words, what is the point of expanding $\phi$ with infinite set $\{I_1, I_2, \ldots\}$ in the proof and applying the Compactness theorem? This says that there is an infinite model. But this is obvious without even without it, right from the the premise of the overspill principle.

What is the point of Löwenheim–Skolem theorem?

Theorem states  

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Edit As sjmc remarks, the statement of the theorem is actually a consecuence of it, known as Overspill Principle. The "Logic in Computer Science" misinforms us!

Nevertheless, I started to understand why people (mistakenly) ask me to differentiate between infinite amount of models and models of infinite size. They fail to recognize that principle "The fact that I can always name a number larger than yours implies that we have an infinite model/set", which is intuitive and used to prove the overspill theorem, also implies that the model of infinite size exists. The set of models $A = \{M_k, M_l, M_m \ldots\}$ of sizes $S = \{k, l, m, \ldots\}$ korrespondingly. When you speak about size of models, you basically speak of the numbers in $S$. When you say "a model of size n" you just say "n". Thus, we can forget about set of models and speak only about S. Now, you say that "whatever integer you have, set S contains a larger one." This basically means that S contains an infinite number (i.e. $A$ contains an enumerable model). What to be proven here?

What is the point of (Compactness theorem in the) Overspill principle?

The principle (called a Löwenheim–Skolem theorem by Huth and Ryan) states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Nevertheless, I started to understand why people (mistakenly) ask me to differentiate between infinite amount of models and models of infinite size. They fail to recognize that principle "The fact that I can always name a number larger than yours implies that we have an infinite model/set", which is intuitive and used to prove the overspill theorem, also implies that the model of infinite size exists. The set of models $A = \{M_k, M_l, M_m \ldots\}$ has sizes $S = \{k, l, m, \ldots\}$ correspondingly. When you speak about size of models, you basically speak of the numbers in $S$. When you say "a model of size n" you just say "n". Thus, we can forget about set of models and speak only about S. Now, you say that "whatever integer you have, set S contains a larger one." This basically means that S contains an infinite number (i.e. $A$ contains infinite models). What to be proven here?

In other words, what is the point of expanding $\phi$ with infinite set $\{I_1, I_2, \ldots\}$ in the proof and applying the Compactness theorem? This says that there is an infinite model. But this is obvious without even without it, right from the the premise of the overspill principle.

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Theorem states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PSPS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

EditEdit As sjmc remarks, the statement of the theorem is actually a consecuence of it, known as Overspill Principle. The "Logic in Computer Science" misinforms us!

Nevertheless, I started to understand why people (mistakenly) ask me to differentiate between infinite amount of models and models of infinite size. They fail to recognize that principle "The fact that I can always name a number larger than yours implies that we have an infinite model/set", which is intuitive and used to prove the overspill theorem, also implies that the model of infinite size exists. The set of models $A = \{M_k, M_l, M_m \ldots\}$ of sizes $S = \{k, l, m, \ldots\}$ korrespondingly. When you speak about size of models, you basically speak of the numbers in $S$. When you say "a model of size n" you just say "n". Thus, we can forget about set of models and speak only about S. Now, you say that "whatever integer you have, set S contains a larger one." This basically means that S contains an infinite number (i.e. $A$ contains an enumerable model). What to be proven here?

Theorem states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Edit As sjmc remarks, the statement of the theorem is actually a consecuence of it, known as Overspill Principle. The "Logic in Computer Science" misinforms us!

Theorem states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Edit As sjmc remarks, the statement of the theorem is actually a consecuence of it, known as Overspill Principle. The "Logic in Computer Science" misinforms us!

Nevertheless, I started to understand why people (mistakenly) ask me to differentiate between infinite amount of models and models of infinite size. They fail to recognize that principle "The fact that I can always name a number larger than yours implies that we have an infinite model/set", which is intuitive and used to prove the overspill theorem, also implies that the model of infinite size exists. The set of models $A = \{M_k, M_l, M_m \ldots\}$ of sizes $S = \{k, l, m, \ldots\}$ korrespondingly. When you speak about size of models, you basically speak of the numbers in $S$. When you say "a model of size n" you just say "n". Thus, we can forget about set of models and speak only about S. Now, you say that "whatever integer you have, set S contains a larger one." This basically means that S contains an infinite number (i.e. $A$ contains an enumerable model). What to be proven here?

4 It is Overspill principle actually
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Theorem statesTheorem states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Edit As sjmc remarks, the statement of the theorem is actually a consecuence of it, known as Overspill Principle. The "Logic in Computer Science" misinforms us!

Theorem states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Theorem states

Let $\phi$ be a sentence of predicate logic such that for any natural number $n \geq 1$, there is a model of $\phi$ with at least $n$ elements. Then $\phi$ has a model with infinitely many elements.

IMO, it basically states that if you can always name a number larger than mine arbitrary natural number then your model is infinite. What needs to be proven here? There are no other options obviously for any school kid.

PS The answers state that there is a difference between having infinite amount of models and single infinite model. But this is similarly stupid. At first, I do not see whether I claim that I have a single infinite model or approach it by having all denumerable models. Secondly, it does not matter since in any case you should have an infinite model in order to respond to any natural number.

Edit As sjmc remarks, the statement of the theorem is actually a consecuence of it, known as Overspill Principle. The "Logic in Computer Science" misinforms us!

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