Define a model for $mathbb N$ without set theory
I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops.
It led me to do a bit of elementary logic and learning what are first-order formal systems, second-order logic, third-order logic, and so on. The notion of a predicate and so on. When that was done, I started having a look at ZF and ZFC, and realized that defining the set $Bbb{N} $ correctly seemed highly non-trivial.
An issue that I have is that the axiom schemas of set theory (such as the axiom schema of Specification or Replacement) use natural numbers in them to admit an arbitrary number of finitely many logical variables in the formula defining the various sets involved. This pre-supposes that there should be a model where natural numbers are defined so that they can be used to construct this theory.
So for me, this meant that there should be a model of the natural numbers that doesn't use set theory, otherwise the axioms of set theory cannot be properly laid down.
Under what theory would such a model be built? The Peano axioms use strongly the notion of a "successor function", which seems to assume the notion of a set already.
natural-numbers
edited 2 days ago
dmtri
1,4301521
asked 2 days ago
Patrick Da Silva
32k354106
|
show 1 more comment
I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops.
It led me to do a bit of elementary logic and learning what are first-order formal systems, second-order logic, third-order logic, and so on. The notion of a predicate and so on. When that was done, I started having a look at ZF and ZFC, and realized that defining the set $Bbb{N} $ correctly seemed highly non-trivial.
An issue that I have is that the axiom schemas of set theory (such as the axiom schema of Specification or Replacement) use natural numbers in them to admit an arbitrary number of finitely many logical variables in the formula defining the various sets involved. This pre-supposes that there should be a model where natural numbers are defined so that they can be used to construct this theory.
So for me, this meant that there should be a model of the natural numbers that doesn't use set theory, otherwise the axioms of set theory cannot be properly laid down.
Under what theory would such a model be built? The Peano axioms use strongly the notion of a "successor function", which seems to assume the notion of a set already.
natural-numbers
edited 2 days ago
dmtri
1,4301521
asked 2 days ago
Patrick Da Silva
32k354106
1
What is your definition of "model"?
– Mark S.
2 days ago
1
Your use of letters, words, sentences..., presupposes the existence of a coherent contextually appropriate interpretation for them.... Just like an axiom schema's use of dummy symbols presupposes a contextually appropriate interpretation for them.
– Not Mike
2 days ago
@MarkS. : An interpretation of a theory that satisfies its axioms. Essentially, maybe I'm misunderstanding how logic is normally done because I'm just beginning to understand formal systems, models, theories... but it seems circular to me that some axioms of set theory look like $forall w_1,cdots,forall w_n$ without even discussing integers. How do we deal with that? Is there some human understanding of the axiom schema involved or is it still mathematical and I'm missing something to work with natural numbers?
– Patrick Da Silva
yesterday
Usually the definition of a model of a first order system is a set, or at least includes a set in a key way, which is why I asked. For instance, see the Wikipedia pages for "interpretation" in logic or model theory. "Some axioms of set theory [involve something like subscripts up to $n$]" Not in ZFC they don't. The closest thing I can think of that comes up would be the concept of a first order formula, but you don't need to be able to count (say, with natural numbers) to build that.
– Mark S.
yesterday
@Mark S. : Then maybe I misunderstood how axioms need to be read. How do I formulate the axiom schema of specification without having the notion of a natural number? It refers to "having an arbitrary but finite number of variables". That is a quite hard to define axiom if you don't know what "finite" means. Where am I wrong?
– Patrick Da Silva
yesterday
|
show 1 more comment
I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops.
It led me to do a bit of elementary logic and learning what are first-order formal systems, second-order logic, third-order logic, and so on. The notion of a predicate and so on. When that was done, I started having a look at ZF and ZFC, and realized that defining the set $Bbb{N} $ correctly seemed highly non-trivial.
An issue that I have is that the axiom schemas of set theory (such as the axiom schema of Specification or Replacement) use natural numbers in them to admit an arbitrary number of finitely many logical variables in the formula defining the various sets involved. This pre-supposes that there should be a model where natural numbers are defined so that they can be used to construct this theory.
So for me, this meant that there should be a model of the natural numbers that doesn't use set theory, otherwise the axioms of set theory cannot be properly laid down.
Under what theory would such a model be built? The Peano axioms use strongly the notion of a "successor function", which seems to assume the notion of a set already.
natural-numbers
edited 2 days ago
dmtri
1,4301521
asked 2 days ago
Patrick Da Silva
32k354106
I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops.
It led me to do a bit of elementary logic and learning what are first-order formal systems, second-order logic, third-order logic, and so on. The notion of a predicate and so on. When that was done, I started having a look at ZF and ZFC, and realized that defining the set $Bbb{N} $ correctly seemed highly non-trivial.
An issue that I have is that the axiom schemas of set theory (such as the axiom schema of Specification or Replacement) use natural numbers in them to admit an arbitrary number of finitely many logical variables in the formula defining the various sets involved. This pre-supposes that there should be a model where natural numbers are defined so that they can be used to construct this theory.
So for me, this meant that there should be a model of the natural numbers that doesn't use set theory, otherwise the axioms of set theory cannot be properly laid down.
Under what theory would such a model be built? The Peano axioms use strongly the notion of a "successor function", which seems to assume the notion of a set already.
natural-numbers
natural-numbers
edited 2 days ago
dmtri
1,4301521
asked 2 days ago
Patrick Da Silva
32k354106
edited 2 days ago
dmtri
1,4301521
asked 2 days ago
Patrick Da Silva
32k354106
edited 2 days ago
dmtri
1,4301521
edited 2 days ago
dmtri
1,4301521
edited 2 days ago
dmtri
1,4301521
1,4301521
asked 2 days ago
Patrick Da Silva
32k354106
asked 2 days ago
Patrick Da Silva
32k354106
asked 2 days ago
Patrick Da Silva
32k354106
32k354106
1
What is your definition of "model"?
– Mark S.
2 days ago
1
Your use of letters, words, sentences..., presupposes the existence of a coherent contextually appropriate interpretation for them.... Just like an axiom schema's use of dummy symbols presupposes a contextually appropriate interpretation for them.
– Not Mike
2 days ago
@MarkS. : An interpretation of a theory that satisfies its axioms. Essentially, maybe I'm misunderstanding how logic is normally done because I'm just beginning to understand formal systems, models, theories... but it seems circular to me that some axioms of set theory look like $forall w_1,cdots,forall w_n$ without even discussing integers. How do we deal with that? Is there some human understanding of the axiom schema involved or is it still mathematical and I'm missing something to work with natural numbers?
– Patrick Da Silva
yesterday
Usually the definition of a model of a first order system is a set, or at least includes a set in a key way, which is why I asked. For instance, see the Wikipedia pages for "interpretation" in logic or model theory. "Some axioms of set theory [involve something like subscripts up to $n$]" Not in ZFC they don't. The closest thing I can think of that comes up would be the concept of a first order formula, but you don't need to be able to count (say, with natural numbers) to build that.
– Mark S.
yesterday
@Mark S. : Then maybe I misunderstood how axioms need to be read. How do I formulate the axiom schema of specification without having the notion of a natural number? It refers to "having an arbitrary but finite number of variables". That is a quite hard to define axiom if you don't know what "finite" means. Where am I wrong?
– Patrick Da Silva
yesterday
|
show 1 more comment
1
What is your definition of "model"?
– Mark S.
2 days ago
1
Your use of letters, words, sentences..., presupposes the existence of a coherent contextually appropriate interpretation for them.... Just like an axiom schema's use of dummy symbols presupposes a contextually appropriate interpretation for them.
– Not Mike
2 days ago
@MarkS. : An interpretation of a theory that satisfies its axioms. Essentially, maybe I'm misunderstanding how logic is normally done because I'm just beginning to understand formal systems, models, theories... but it seems circular to me that some axioms of set theory look like $forall w_1,cdots,forall w_n$ without even discussing integers. How do we deal with that? Is there some human understanding of the axiom schema involved or is it still mathematical and I'm missing something to work with natural numbers?
– Patrick Da Silva
yesterday
Usually the definition of a model of a first order system is a set, or at least includes a set in a key way, which is why I asked. For instance, see the Wikipedia pages for "interpretation" in logic or model theory. "Some axioms of set theory [involve something like subscripts up to $n$]" Not in ZFC they don't. The closest thing I can think of that comes up would be the concept of a first order formula, but you don't need to be able to count (say, with natural numbers) to build that.
– Mark S.
yesterday
@Mark S. : Then maybe I misunderstood how axioms need to be read. How do I formulate the axiom schema of specification without having the notion of a natural number? It refers to "having an arbitrary but finite number of variables". That is a quite hard to define axiom if you don't know what "finite" means. Where am I wrong?
– Patrick Da Silva
yesterday
1
1
What is your definition of "model"?
– Mark S.
2 days ago
What is your definition of "model"?
– Mark S.
2 days ago
1
1
Your use of letters, words, sentences..., presupposes the existence of a coherent contextually appropriate interpretation for them.... Just like an axiom schema's use of dummy symbols presupposes a contextually appropriate interpretation for them.
– Not Mike
2 days ago
Your use of letters, words, sentences..., presupposes the existence of a coherent contextually appropriate interpretation for them.... Just like an axiom schema's use of dummy symbols presupposes a contextually appropriate interpretation for them.
– Not Mike
2 days ago
@MarkS. : An interpretation of a theory that satisfies its axioms. Essentially, maybe I'm misunderstanding how logic is normally done because I'm just beginning to understand formal systems, models, theories... but it seems circular to me that some axioms of set theory look like $forall w_1,cdots,forall w_n$ without even discussing integers. How do we deal with that? Is there some human understanding of the axiom schema involved or is it still mathematical and I'm missing something to work with natural numbers?
– Patrick Da Silva
yesterday
@MarkS. : An interpretation of a theory that satisfies its axioms. Essentially, maybe I'm misunderstanding how logic is normally done because I'm just beginning to understand formal systems, models, theories... but it seems circular to me that some axioms of set theory look like $forall w_1,cdots,forall w_n$ without even discussing integers. How do we deal with that? Is there some human understanding of the axiom schema involved or is it still mathematical and I'm missing something to work with natural numbers?
– Patrick Da Silva
yesterday
Usually the definition of a model of a first order system is a set, or at least includes a set in a key way, which is why I asked. For instance, see the Wikipedia pages for "interpretation" in logic or model theory. "Some axioms of set theory [involve something like subscripts up to $n$]" Not in ZFC they don't. The closest thing I can think of that comes up would be the concept of a first order formula, but you don't need to be able to count (say, with natural numbers) to build that.
– Mark S.
yesterday
Usually the definition of a model of a first order system is a set, or at least includes a set in a key way, which is why I asked. For instance, see the Wikipedia pages for "interpretation" in logic or model theory. "Some axioms of set theory [involve something like subscripts up to $n$]" Not in ZFC they don't. The closest thing I can think of that comes up would be the concept of a first order formula, but you don't need to be able to count (say, with natural numbers) to build that.
– Mark S.
yesterday
@Mark S. : Then maybe I misunderstood how axioms need to be read. How do I formulate the axiom schema of specification without having the notion of a natural number? It refers to "having an arbitrary but finite number of variables". That is a quite hard to define axiom if you don't know what "finite" means. Where am I wrong?
– Patrick Da Silva
yesterday
@Mark S. : Then maybe I misunderstood how axioms need to be read. How do I formulate the axiom schema of specification without having the notion of a natural number? It refers to "having an arbitrary but finite number of variables". That is a quite hard to define axiom if you don't know what "finite" means. Where am I wrong?
– Patrick Da Silva
yesterday
|
show 1 more comment
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1
What is your definition of "model"?
– Mark S.
2 days ago
1
Your use of letters, words, sentences..., presupposes the existence of a coherent contextually appropriate interpretation for them.... Just like an axiom schema's use of dummy symbols presupposes a contextually appropriate interpretation for them.
– Not Mike
2 days ago
@MarkS. : An interpretation of a theory that satisfies its axioms. Essentially, maybe I'm misunderstanding how logic is normally done because I'm just beginning to understand formal systems, models, theories... but it seems circular to me that some axioms of set theory look like $forall w_1,cdots,forall w_n$ without even discussing integers. How do we deal with that? Is there some human understanding of the axiom schema involved or is it still mathematical and I'm missing something to work with natural numbers?
– Patrick Da Silva
yesterday
Usually the definition of a model of a first order system is a set, or at least includes a set in a key way, which is why I asked. For instance, see the Wikipedia pages for "interpretation" in logic or model theory. "Some axioms of set theory [involve something like subscripts up to $n$]" Not in ZFC they don't. The closest thing I can think of that comes up would be the concept of a first order formula, but you don't need to be able to count (say, with natural numbers) to build that.
– Mark S.
yesterday
@Mark S. : Then maybe I misunderstood how axioms need to be read. How do I formulate the axiom schema of specification without having the notion of a natural number? It refers to "having an arbitrary but finite number of variables". That is a quite hard to define axiom if you don't know what "finite" means. Where am I wrong?
– Patrick Da Silva
yesterday