如何用标准数学符号表示‘存在无穷多个整数n’？

Great question! Notation consistency is super important in math, so let's break this down clearly:

First up: the two notations you proposed — $\exists \text{ infinite } n \in\mathbb{Z}$ and $\exists\text{ inf }n\in\mathbb{Z}$ — aren't standard mathematical usage. While a reader might guess what you mean, you won't see these in formal papers, textbooks, or rigorous proofs. They're too informal and don't align with widely accepted conventions.

Now, here are the standard ways to express "there exist infinitely many integers n":

1. The most common shorthand #

A widely recognized shorthand in mathematical literature is $\exists^\infty n \in \mathbb{Z}$. It's concise and immediately clear to most mathematicians. For example, to say "there exist infinitely many even integers", you'd write:
$$\exists^\infty n \in \mathbb{Z}, 2 \mid n$$

2. Using set cardinality #

For a more rigorous, explicit statement, you can use the concept of set cardinality (the number of elements in a set):
$$\left| { n \in \mathbb{Z} \mid P(n) } \right| = \aleph_0$$
Here, $\aleph_0$ is the cardinality of countable infinite sets (like the integers themselves, so any infinite subset of $\mathbb{Z}$ will also have this cardinality). $P(n)$ stands for the proposition about n you're describing. Using the even integers example again:
$$\left| { n \in \mathbb{Z} \mid 2 \mid n } \right| = \aleph_0$$
This is a very precise way to state the idea, especially if you need to emphasize the "infinite" aspect's formal set-theoretic meaning.

3. Formal first-order logic #

If you need to express this purely in first-order logic (it's more verbose, but useful for formal proofs), you can write:
$$\forall k \in \mathbb{N}, \exists n_1, n_2, \dots, n_k \in \mathbb{Z}, \left( \bigwedge_{1 \leq i < j \leq k} n_i \neq n_j \right) \land \left( \bigwedge_{1 \leq i \leq k} P(n_i) \right)$$
In plain language, this means "for any positive integer k, you can find k distinct integers n where each n satisfies the proposition P(n)" — which logically equates to "there are infinitely many such n".

As for your creative idea: $\operatorname*{\text{Exi}}\limits_{n\to\infty}n\in\mathbb{Z}$ — it's a clever take inspired by limit notation, but it's not part of the standard mathematical symbol system. Other mathematicians might be confused, since $\lim_{n\to\infty}$ is tied to limit operations, not existential quantifiers. Stick to the standard options above for clarity.

内容的提问来源于stack exchange，提问作者Mr Pie