I've been exploring a very simple variation of Zeno’s paradox and would like to share it here for feedback — especially from those with a background in constructive mathematics, logic, or analysis.
Setup: We project natural numbers N\mathbb{N}N one by one into the interval [0,1)[0,1)[0,1). The rule is:
- Place the first number at 0.5,
- The second at 0.75,
- The third at 0.875,
- And so on — each new number goes halfway between 1 and the current rightmost point.
But here’s the twist: Every time a new number is added, all previous points are proportionally compressed to fit inside the interval [0,pk)[0, p_k)[0,pk), where pkp_kpk is the position of the newly added number. This simulates a constructive process: only finite data is used at each stage, and no infinite set is ever handled directly.
The paradox arises in the limit.
Let me explain the contradiction in two parts:
1. Every number ends up near 0 — unavoidably.
For any fixed nnn, its projected position behaves roughly like nN\frac{n}{N}Nn when there are NNN numbers placed. So, as N→∞N \to \inftyN→∞, the ratio nN→0\frac{n}{N} \to 0Nn→0. This means that:
- No matter how large nnn is,
- And no matter how we divide [0,1)[0,1)[0,1) into kkk equal subintervals,
- Eventually nN
Yes, constructive processes exist that:
Generate sequences dense in
[
0
,
1
)
[0,1)
Can be structured so that values (even indexed by
𝑛
n) approach 0 arbitrarily closely
Can resemble Zeno's paradox (e.g., Van der Corput sequence, 1/n sequences, or modular inverses)
If your interest is in constructive dynamical systems, symbolic dynamics, or ergodic theory, this question overlaps deeply with those fields.
es, a Zeno-like constructive process can densely fill the interval [0,1)[0,1)[0,1), but pushing every natural number nnn arbitrarily close to 0 is not possible in the standard sense, because:
- Natural numbers n∈Nn \in \mathbb{N}n∈N are discrete, while [0,1)[0,1)[0,1) is dense and continuous.
- You cannot "move" discrete integers into [0,1)[0,1)[0,1) directly—unless you map or encode them via a function or transformation.
1) If you mean a single sequence (an)⊂[0,1)(a_n)\subset[0,1)(an)⊂[0,1) that is dense in [0,1)[0,1)[0,1) and satisfies an→0a_n\to0an→0: impossible. A sequence that converges to 0 cannot be dense in [0,1)[0,1)[0,1) (points arbitrarily close to 1 cannot occur eventually).
2) If you mean a constructive process that assigns to each natural number nnn a sequence of positions xn(1),xn(2),…x_n(1),x_n(2),\dotsxn(1),xn(2),… in [0,1)[0,1)[0,1) (so every nnn has its own trajectory), and you only require that for each fixed nnn the trajectory comes arbitrarily close to 0 infinitely often while the union of all visited points is dense: yes, possible.
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