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Galileo’s Paradox Finite Demonstration — result sheet
Count positive integers and perfect squares up to a finite limit to demonstrate the pairing behind Galileo’s infinity paradox without pretending to measure infinity.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: Positive integers through N = N; perfect squares through N = floor(√N); non-squares = N − floor(√N); finite square share = floor(√N) ÷ N × 100.
Galileo’s example pairs each positive integer n with its square n². This finite calculator shows how many squares appear below a chosen limit and how sparse they look in that finite window. The pairing itself is the important idea: an infinite set can be put in one-to-one correspondence with a proper subset, so a finite percentage is not an answer to the size of an infinite set.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Finite demonstration limit — positive integers; minimum 1; maximum 1000000000
Worked example
| Input | Value |
|---|---|
| Finite demonstration limit | 1000 |
There are 1,000 positive integers through 1,000, 31 perfect squares, 969 non-squares, and a finite square share of 3.1%.
Assumptions and limits
- The limit is a finite positive integer and the count includes integers 1 through the limit.
- A perfect square is counted when n² is at most the finite limit.
- The displayed square share describes only the selected finite window and changes as the window grows.
- The calculator does not assign a percentage or ordinary size to an actually infinite set.
- The pairing n→n² is one-to-one over positive integers but is shown here through finite counts only.
- This is an educational demonstration, not a probability model, limit proof, or cardinal-arithmetic solver.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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