MHB Can we take limits to infinity in finite sets of $\Bbb{N}$?

AI Thread Summary
In the discussion, participants explore whether limits can be taken to infinity within finite sets of natural numbers, concluding that this concept does not apply to finite sets. They provide examples of infinite subsets of natural numbers, such as even numbers (2ℕ) and odd numbers (2ℕ + 1). Additionally, the set of prime numbers is mentioned as another example of an infinite subset. The conversation emphasizes the distinction between finite and infinite sets in the context of limits. Overall, the topic centers on the properties of finite versus infinite sets in relation to limits.
ozkan12
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İn a finite set, can we take limit to $\infty$ ?

Also, can you give an example related to infinite subset of $\Bbb{N}$ ?
 
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ozkan12 said:
İn a finite set, can we take limit to $\infty$ ?

Also, can you give an example related to infinite subset of $\Bbb{N}$ ?

The first question is not clear. For the second one, $2 \Bbb{N}$ which constitutes even integers is a an infinite subset.
 
Dear ZaidAlyafey,

Thank you for your attention...For second question odd numbers can be an example...İs there any examples anything else 2N and 2N+1
 
ozkan12 said:
Dear ZaidAlyafey,

Thank you for your attention...For second question odd numbers can be an example...İs there any examples anything else 2N and 2N+1

Yes. For example, the set of prime numbers is infinite. More generally, any sequence $\{a_i\}^{\infty}_1$ where $a_i \in \mathbb{N}$ is an infinite subset of natural numbers.
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...

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