Term structure isomorphic to the usual model/structure of number theory

In summary, the question is whether the Henkin structure associated to \mathrm{Th}(\mathbb{N}) can be (isomorphic to) \mathbb{N}, and the answer is yes. This is achieved by adding \omega constants and axioms to the language and theory, and then extending to a complete theory and constructing the model as the set of variable free terms. By properly interpreting the Henkin constants and axioms, the resulting structure will be isomorphic to \mathbb{N}.
  • #1
wisvuze
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Hello, suppose I have a set of sentences Ʃ from the language of number theory ( the usual one ). Then, I extend this to a maximally consistent set of sentences Ʃ' and create a henkin term structure for it ( i.e. as in the popular proof of the completeness theorem ). Can it be true that this resulting structure is isomorphic to the standard structure/model of number theory? Usually, it isn't enough for two structures to satisfy the same sentences for them to be isomorphic, so I am not sure..

thanks
 
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  • #2
Although two structures satisfying the same sentences (i.e. being elementarily equivalent) is not sufficient for them to be isomorphic, it is necessary, thus to answer your question: Can the Henkin structure associated to [itex]\Sigma'[/itex] be isomorphic to [itex]\mathbb{N}[/itex]? it would be necessary for [itex]\Sigma' = \mathrm{Th}(\mathbb{N})[/itex], the full theory of [itex]\mathbb{N}[/itex].

In other words, the question is whether the Henkin structure associated to [itex]\mathrm{Th}(\mathbb{N})[/itex] can be (isomorphic to) [itex]\mathbb{N}[/itex]. The answer is Yes. In the Henkin construction, you add [itex]\omega[/itex] constants to your language and [itex]\omega[/itex] axioms to your theory. You then repeat this [itex]\omega[/itex] times. You then extend to a complete theory. And then you construct the model as the set of variable free terms in your new language, modulo being provably equivalent by your new theory. What you need to do is make sure that when extending to a complete theory, for every Henkin constant c there is some "SS...S0" such that "c = SS...S0" is added to your theory.

This is easy: Look a the first set of Henkin axioms you added, they're of the form [itex]\exists x \phi (x) \rightarrow \phi (c)[/itex]. If [itex]\exists x\phi (x)[/itex] is true in [itex]\mathbb{N}[/itex], say [itex]n[/itex] is the minimal witness, then add [itex]c = \bar{n}[/itex] to your theory in the final extend-to-a-complete-theory stage. Here [itex]\bar{n}[/itex] is [itex]n[/itex] S's, followed by a 0 symbol. If [itex]\exists x\phi(x)[/itex] doesn't hold, add [itex]c = 0[/itex]. Now deal with the Henkin axioms added in the second iteration in the same way (by looking at whether the existential sentence is true in [itex]\mathbb{N}[/itex]), interpreting any occurrence of the first set of Henkin constants according to the interpretation we just fixed above. Do this for all the Henkin constants/axioms, and then complete the theory as usual.

It's not hard to see that this will be isomorphic to [itex]\mathbb{N}[/itex].
 
  • #3
Thanks for the reply, I appreciate it
 

1. What is the term structure isomorphic to the usual model/structure of number theory?

The term structure isomorphic to the usual model/structure of number theory refers to a mathematical structure that is equivalent to the standard model of number theory. This means that the two structures have the same properties and axioms, and any statement that is true in one structure will also be true in the other.

2. How is the term structure isomorphic to the usual model/structure of number theory useful in mathematics?

The term structure isomorphic to the usual model/structure of number theory is useful in mathematics because it allows for the study of number theory using different mathematical structures. This can provide new insights and perspectives on the subject and can also aid in solving complex problems.

3. What are some examples of structures that are isomorphic to the usual model of number theory?

Some examples of structures that are isomorphic to the usual model of number theory include the p-adic numbers, the algebraic closure of the field of rational numbers, and the ring of algebraic integers.

4. How is the term structure isomorphic to the usual model/structure of number theory related to other branches of mathematics?

The term structure isomorphic to the usual model/structure of number theory is closely related to other branches of mathematics, such as algebra, geometry, and topology. This is because many mathematical structures can be viewed as special cases of the structure of number theory.

5. Can the term structure isomorphic to the usual model/structure of number theory be applied in other fields besides mathematics?

Yes, the term structure isomorphic to the usual model/structure of number theory can be applied in other fields besides mathematics. For example, it has been used in computer science and cryptography to develop efficient algorithms and secure encryption methods based on number theory concepts.

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