Sentence over the empty language saying a model is infinite
$begingroup$
So the question is whether there is a sentence $phi$ over the empty language such that for any structure $M$, $Mmodelsphi$ iff $M$ is infinite.
I’m pretty sure the answer is no, but I don’t see why. Compactness seems relevant, but I don’t see how.
model-theory
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
$endgroup$
add a comment |
$begingroup$
So the question is whether there is a sentence $phi$ over the empty language such that for any structure $M$, $Mmodelsphi$ iff $M$ is infinite.
I’m pretty sure the answer is no, but I don’t see why. Compactness seems relevant, but I don’t see how.
model-theory
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
$endgroup$
add a comment |
$begingroup$
So the question is whether there is a sentence $phi$ over the empty language such that for any structure $M$, $Mmodelsphi$ iff $M$ is infinite.
I’m pretty sure the answer is no, but I don’t see why. Compactness seems relevant, but I don’t see how.
model-theory
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
$endgroup$
So the question is whether there is a sentence $phi$ over the empty language such that for any structure $M$, $Mmodelsphi$ iff $M$ is infinite.
I’m pretty sure the answer is no, but I don’t see why. Compactness seems relevant, but I don’t see how.
model-theory
model-theory
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
asked Dec 14 '18 at 20:36
ReveillarkReveillark
4,662822
4,662822
add a comment |
add a comment |
1 Answer
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$begingroup$
HINT: consider its negation. This is a sentence satisfied in precisely the finite structures.
Now, there is a standard corollary of compactness: if $T$ has models of arbitrarily large finite size, what can you conclude?
We can also avoid compactness here: via Ehrenfeucht-Fraisse games, you can show that every satisfiable sentence in the empty theory has a finite model, and indeed we can effectively find a bound on how large this model has to be. For example, if $varphi$ has the form $exists x_1,..., x_n[quantifierfree]$, ...
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
$endgroup$
$begingroup$
So the theory consisting of just the negation of that sentence has models of arbitrarily large finite size, thus has an infinite model by compactness, contradiction. Thanks!
$endgroup$
– Reveillark
Dec 14 '18 at 20:45
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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$begingroup$
HINT: consider its negation. This is a sentence satisfied in precisely the finite structures.
Now, there is a standard corollary of compactness: if $T$ has models of arbitrarily large finite size, what can you conclude?
We can also avoid compactness here: via Ehrenfeucht-Fraisse games, you can show that every satisfiable sentence in the empty theory has a finite model, and indeed we can effectively find a bound on how large this model has to be. For example, if $varphi$ has the form $exists x_1,..., x_n[quantifierfree]$, ...
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
$endgroup$
$begingroup$
So the theory consisting of just the negation of that sentence has models of arbitrarily large finite size, thus has an infinite model by compactness, contradiction. Thanks!
$endgroup$
– Reveillark
Dec 14 '18 at 20:45
add a comment |
$begingroup$
HINT: consider its negation. This is a sentence satisfied in precisely the finite structures.
Now, there is a standard corollary of compactness: if $T$ has models of arbitrarily large finite size, what can you conclude?
We can also avoid compactness here: via Ehrenfeucht-Fraisse games, you can show that every satisfiable sentence in the empty theory has a finite model, and indeed we can effectively find a bound on how large this model has to be. For example, if $varphi$ has the form $exists x_1,..., x_n[quantifierfree]$, ...
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
$endgroup$
$begingroup$
So the theory consisting of just the negation of that sentence has models of arbitrarily large finite size, thus has an infinite model by compactness, contradiction. Thanks!
$endgroup$
– Reveillark
Dec 14 '18 at 20:45
add a comment |
$begingroup$
HINT: consider its negation. This is a sentence satisfied in precisely the finite structures.
Now, there is a standard corollary of compactness: if $T$ has models of arbitrarily large finite size, what can you conclude?
We can also avoid compactness here: via Ehrenfeucht-Fraisse games, you can show that every satisfiable sentence in the empty theory has a finite model, and indeed we can effectively find a bound on how large this model has to be. For example, if $varphi$ has the form $exists x_1,..., x_n[quantifierfree]$, ...
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
$endgroup$
HINT: consider its negation. This is a sentence satisfied in precisely the finite structures.
Now, there is a standard corollary of compactness: if $T$ has models of arbitrarily large finite size, what can you conclude?
We can also avoid compactness here: via Ehrenfeucht-Fraisse games, you can show that every satisfiable sentence in the empty theory has a finite model, and indeed we can effectively find a bound on how large this model has to be. For example, if $varphi$ has the form $exists x_1,..., x_n[quantifierfree]$, ...
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
edited Dec 14 '18 at 20:46
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
answered Dec 14 '18 at 20:43
Noah SchweberNoah Schweber
126k10151290
126k10151290
$begingroup$
So the theory consisting of just the negation of that sentence has models of arbitrarily large finite size, thus has an infinite model by compactness, contradiction. Thanks!
$endgroup$
– Reveillark
Dec 14 '18 at 20:45
add a comment |
$begingroup$
So the theory consisting of just the negation of that sentence has models of arbitrarily large finite size, thus has an infinite model by compactness, contradiction. Thanks!
$endgroup$
– Reveillark
Dec 14 '18 at 20:45
$begingroup$
So the theory consisting of just the negation of that sentence has models of arbitrarily large finite size, thus has an infinite model by compactness, contradiction. Thanks!
$endgroup$
– Reveillark
Dec 14 '18 at 20:45
$begingroup$
So the theory consisting of just the negation of that sentence has models of arbitrarily large finite size, thus has an infinite model by compactness, contradiction. Thanks!
$endgroup$
– Reveillark
Dec 14 '18 at 20:45
add a comment |
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