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I think that you mean by feel your construction non constructive? because you have no way to establish a priori if a given real number is rational or irrational. The following construction avoids the problem: I suppose directly that $X=\mathbb N$. For any $\frac{1}{10}\leq t< 1$, for instance $t=0,324145...$, define $I_t$ to be the set containing the following natural numbers

$$3,32,324,3241,32414,324145,\ldots$$

The family $I_t$ is uncountable and $|I_t\cap I_s|<\infty$, for all $t\neq s$.

This looks quite similar to your construction, but it sounds more constructive in the sense that you do not need to know a priori that $t$ is irrational. Indeed it works anyway.

1

What do you mean by constructive? I suppose directly that $X=\mathbb N$. For any $\frac{1}{10}\leq t< 1$, for instance $t=0,324145...$, define $I_t$ to be the set containing the following natural numbers

$$3,32,324,3241,32414,324145,\ldots$$

The family $I_t$ is uncountable and $|I_t\cap I_s|<\infty$, for all $t\neq s$.

This looks quite similar to your construction, but it sounds more constructive in the sense that you do not need to know a priori that $t$ is irrational. Indeed it works anyway.