Generalizing the construction of $PA^omega$
I'm taking an extremely basic course in mathematical logic where we briefly talked about nonstandard models of arithmetic. The only example we worked through has been the construction of $PA^omega$ through the compactness theorem. EDIT: The axioms of $PA^omega$ consist of the axioms of $PA$ along an infinite set of axioms of the form $omega > n$ for each standard natural number $n$ (the notation is kind of unfortunate, perhaps I should have chosen a better name than $omega$ for this symbol...).
I was wondering can this process of "extending" $PA$ to $PA^omega$ be generalized?
I thought of defining something like $(PA^omega)^{omega'}$ adding more axioms of the form $omega ' > n$ for $n$ in $PA^omega$ but I'm not even sure this "in" would be well-defined. Is there a way to define such a model?
logic nonstandard-models
asked Dec 4 '18 at 16:43
Ettore
9518
add a comment |
I'm taking an extremely basic course in mathematical logic where we briefly talked about nonstandard models of arithmetic. The only example we worked through has been the construction of $PA^omega$ through the compactness theorem. EDIT: The axioms of $PA^omega$ consist of the axioms of $PA$ along an infinite set of axioms of the form $omega > n$ for each standard natural number $n$ (the notation is kind of unfortunate, perhaps I should have chosen a better name than $omega$ for this symbol...).
I was wondering can this process of "extending" $PA$ to $PA^omega$ be generalized?
I thought of defining something like $(PA^omega)^{omega'}$ adding more axioms of the form $omega ' > n$ for $n$ in $PA^omega$ but I'm not even sure this "in" would be well-defined. Is there a way to define such a model?
logic nonstandard-models
asked Dec 4 '18 at 16:43
Ettore
9518
Please define what precisely you mean by $PA^omega$, it is not notation I immediately recognize.
– Andrés E. Caicedo
Dec 4 '18 at 18:34
@AndrésE.Caicedo I added more details. Is there a more commonly accepted name for this model?
– Ettore
Dec 4 '18 at 19:06
2
@Ettore You seem to be confused about the distinction between models and theories. $text{PA}^omega$ is a theory, which has many models. Similarly, you ask "is there a way to define such a model", but it looks like you're trying to describe a theory.
– Alex Kruckman
Dec 4 '18 at 23:13
@AlexKruckman I think you're right regarding both my confusion and my question, the part I'm interested in is actually the construction of such a theory. I don't know if this question is salvageable then, I'll think more about it...
– Ettore
Dec 5 '18 at 10:36
add a comment |
I'm taking an extremely basic course in mathematical logic where we briefly talked about nonstandard models of arithmetic. The only example we worked through has been the construction of $PA^omega$ through the compactness theorem. EDIT: The axioms of $PA^omega$ consist of the axioms of $PA$ along an infinite set of axioms of the form $omega > n$ for each standard natural number $n$ (the notation is kind of unfortunate, perhaps I should have chosen a better name than $omega$ for this symbol...).
I was wondering can this process of "extending" $PA$ to $PA^omega$ be generalized?
I thought of defining something like $(PA^omega)^{omega'}$ adding more axioms of the form $omega ' > n$ for $n$ in $PA^omega$ but I'm not even sure this "in" would be well-defined. Is there a way to define such a model?
logic nonstandard-models
asked Dec 4 '18 at 16:43
Ettore
9518
I'm taking an extremely basic course in mathematical logic where we briefly talked about nonstandard models of arithmetic. The only example we worked through has been the construction of $PA^omega$ through the compactness theorem. EDIT: The axioms of $PA^omega$ consist of the axioms of $PA$ along an infinite set of axioms of the form $omega > n$ for each standard natural number $n$ (the notation is kind of unfortunate, perhaps I should have chosen a better name than $omega$ for this symbol...).
I was wondering can this process of "extending" $PA$ to $PA^omega$ be generalized?
I thought of defining something like $(PA^omega)^{omega'}$ adding more axioms of the form $omega ' > n$ for $n$ in $PA^omega$ but I'm not even sure this "in" would be well-defined. Is there a way to define such a model?
logic nonstandard-models
logic nonstandard-models
asked Dec 4 '18 at 16:43
Ettore
9518
asked Dec 4 '18 at 16:43
Ettore
9518
edited Dec 4 '18 at 20:36
asked Dec 4 '18 at 16:43
Ettore
9518
asked Dec 4 '18 at 16:43
Ettore
9518
asked Dec 4 '18 at 16:43
Ettore
9518
9518
Please define what precisely you mean by $PA^omega$, it is not notation I immediately recognize.
– Andrés E. Caicedo
Dec 4 '18 at 18:34
@AndrésE.Caicedo I added more details. Is there a more commonly accepted name for this model?
– Ettore
Dec 4 '18 at 19:06
2
@Ettore You seem to be confused about the distinction between models and theories. $text{PA}^omega$ is a theory, which has many models. Similarly, you ask "is there a way to define such a model", but it looks like you're trying to describe a theory.
– Alex Kruckman
Dec 4 '18 at 23:13
@AlexKruckman I think you're right regarding both my confusion and my question, the part I'm interested in is actually the construction of such a theory. I don't know if this question is salvageable then, I'll think more about it...
– Ettore
Dec 5 '18 at 10:36
add a comment |
Please define what precisely you mean by $PA^omega$, it is not notation I immediately recognize.
– Andrés E. Caicedo
Dec 4 '18 at 18:34
@AndrésE.Caicedo I added more details. Is there a more commonly accepted name for this model?
– Ettore
Dec 4 '18 at 19:06
2
@Ettore You seem to be confused about the distinction between models and theories. $text{PA}^omega$ is a theory, which has many models. Similarly, you ask "is there a way to define such a model", but it looks like you're trying to describe a theory.
– Alex Kruckman
Dec 4 '18 at 23:13
@AlexKruckman I think you're right regarding both my confusion and my question, the part I'm interested in is actually the construction of such a theory. I don't know if this question is salvageable then, I'll think more about it...
– Ettore
Dec 5 '18 at 10:36
Please define what precisely you mean by $PA^omega$, it is not notation I immediately recognize.
– Andrés E. Caicedo
Dec 4 '18 at 18:34
Please define what precisely you mean by $PA^omega$, it is not notation I immediately recognize.
– Andrés E. Caicedo
Dec 4 '18 at 18:34
@AndrésE.Caicedo I added more details. Is there a more commonly accepted name for this model?
– Ettore
Dec 4 '18 at 19:06
@AndrésE.Caicedo I added more details. Is there a more commonly accepted name for this model?
– Ettore
Dec 4 '18 at 19:06
2
2
@Ettore You seem to be confused about the distinction between models and theories. $text{PA}^omega$ is a theory, which has many models. Similarly, you ask "is there a way to define such a model", but it looks like you're trying to describe a theory.
– Alex Kruckman
Dec 4 '18 at 23:13
@Ettore You seem to be confused about the distinction between models and theories. $text{PA}^omega$ is a theory, which has many models. Similarly, you ask "is there a way to define such a model", but it looks like you're trying to describe a theory.
– Alex Kruckman
Dec 4 '18 at 23:13
@AlexKruckman I think you're right regarding both my confusion and my question, the part I'm interested in is actually the construction of such a theory. I don't know if this question is salvageable then, I'll think more about it...
– Ettore
Dec 5 '18 at 10:36
@AlexKruckman I think you're right regarding both my confusion and my question, the part I'm interested in is actually the construction of such a theory. I don't know if this question is salvageable then, I'll think more about it...
– Ettore
Dec 5 '18 at 10:36
add a comment |
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Please define what precisely you mean by $PA^omega$, it is not notation I immediately recognize.
– Andrés E. Caicedo
Dec 4 '18 at 18:34
@AndrésE.Caicedo I added more details. Is there a more commonly accepted name for this model?
– Ettore
Dec 4 '18 at 19:06
2
@Ettore You seem to be confused about the distinction between models and theories. $text{PA}^omega$ is a theory, which has many models. Similarly, you ask "is there a way to define such a model", but it looks like you're trying to describe a theory.
– Alex Kruckman
Dec 4 '18 at 23:13
@AlexKruckman I think you're right regarding both my confusion and my question, the part I'm interested in is actually the construction of such a theory. I don't know if this question is salvageable then, I'll think more about it...
– Ettore
Dec 5 '18 at 10:36