3 edited body; edited tags

This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here.

From GodelGödel completeness and incompleteness theorems, assuming the standard axioms of math being consistent, the set of natural numbers N could look very different. However, in some sense, there seems to be the one we prefer, namely, the one that consists of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard, so to speak.

However, in the absence of an external world that allows us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true set of natural numbers? What are some of the known arguments for and against there being only one true set of natural numbers?

Related question At what order of logic do we have a unique model of the natural numbers? addresses a similar issue, but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.

This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here.

From Godel completeness and incompleteness theorems, assuming the standard axioms of math being consistent, the set of natural numbers N could look very different. However, in some sense, there seems to be the one we prefer, namely, the one that consists of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard, so to speak.

However, in the absence of an external world that allows us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true set of natural numbers? What are some of the known arguments for and against there being only one true set of natural numbers?

Related question At what order of logic do we have a unique model of the natural numbers? addresses a similar issue, but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.

This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here.

From Gödel completeness and incompleteness theorems, assuming the standard axioms of math being consistent, the set of natural numbers N could look very different. However, in some sense, there seems to be the one we prefer, namely, the one that consists of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard, so to speak.

However, in the absence of an external world that allows us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true set of natural numbers? What are some of the known arguments for and against there being only one true set of natural numbers?

Related question At what order of logic do we have a unique model of the natural numbers? addresses a similar issue, but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.

2 added 62 characters in body; edited tags; edited title

# What are the arguments for and against one"one true natural numberarithmetic"?

This question was born out of a discussion here https://math.stackexchange.com/questions/2246085/is-the-real-number-structure-uniqueIs the real number structure unique? but on Math SE, but since it is more philosophical than mathematicsmathematical I decided to ask here.

From Godel completeness theorem and incompleteness theoremtheorems, assuming the standard axioms of mathsmath being consistent, the set of natural numbernumbers N could look very different. However, in some sense, there seemedseems to be the one we prefer, namely, the one that consistconsists of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard, so to speak.

However, in the absence of an external world that allowallows us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true set of natural numbernumbers? What are some of the known argumentarguments for and against there being only one true set of natural numbers?

Thank you.

Related question: At what order of logic do we have a unique model of the natural numbers? but addresses a similar issue, but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.

# What are the arguments for and against one true natural number?

This question was born out of a discussion here https://math.stackexchange.com/questions/2246085/is-the-real-number-structure-unique but since it is more philosophical than mathematics I decided to ask here.

From Godel completeness theorem and incompleteness theorem, assuming the standard axioms of maths being consistent, the set of natural number N could look very different. However, in some sense, there seemed to be the one we prefer, namely, the one that consist of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard so to speak.

However, in the absence of an external world that allow us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true natural number? What are some of the known argument for and against there being only one true set of natural numbers?

Thank you.

Related question: At what order of logic do we have a unique model of the natural numbers? but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.

# What are the arguments for and against "one true arithmetic"?

This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here.

From Godel completeness and incompleteness theorems, assuming the standard axioms of math being consistent, the set of natural numbers N could look very different. However, in some sense, there seems to be the one we prefer, namely, the one that consists of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard, so to speak.

However, in the absence of an external world that allows us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true set of natural numbers? What are some of the known arguments for and against there being only one true set of natural numbers?

Related question At what order of logic do we have a unique model of the natural numbers? addresses a similar issue, but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.

1

# What are the arguments for and against one true natural number?

This question was born out of a discussion here https://math.stackexchange.com/questions/2246085/is-the-real-number-structure-unique but since it is more philosophical than mathematics I decided to ask here.

From Godel completeness theorem and incompleteness theorem, assuming the standard axioms of maths being consistent, the set of natural number N could look very different. However, in some sense, there seemed to be the one we prefer, namely, the one that consist of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard so to speak.

However, in the absence of an external world that allow us to define rigorously the meaning of "repeatedly adding 1 together", is there in some sense one true natural number? What are some of the known argument for and against there being only one true set of natural numbers?

Thank you.

Related question: At what order of logic do we have a unique model of the natural numbers? but note that this one use second-order arithmetic, which is dependent on some sort of meta-theory in the external world.