Dear Enrico P. G. Cadeddu,
Your comment presents an inconsistent proposition because it appears to contradict the fundamental nature of infinity as defined in mathematics.
Infinity is Unreachable in a Finite Sense:
Infinity, by definition, is not something that can be "reached" or "constructed" in a stepwise manner from finite elements. It exists as a concept beyond any finite bounds, whether represented through numbers, sets, or sequences.
Proper Subsets of an Infinite Set Do Not Dictate Its Infinite Character:
An infinite set remains infinite regardless of the nature of its proper subsets.
Some proper subsets can be finite e.g., {1,2,3} ⊂ N, while others can be infinite e.g., the set of even numbers within N.
The union of infinite subsets can still be infinite, so claiming that a union of proper subsets results in something "not infinite" suggests a misunderstanding of set theory.
Infinity as a Defined Mathematical Concept is Self-Consistent:
The Peano axioms and the axiom of infinity in set theory define an internally consistent framework for handling infinite sets like N.
Any argument that rejects infinity yet still relies on the structure of N (which is inherently infinite) creates a paradox.
Conclusion:
The claim in your text only holds if one assumes an inconsistent mathematical principle, which contradicts established definitions.
The very nature of an infinite set remains infinite, and its proper subsets (whether finite or infinite) do not alter its infinite character.
Infinity is not something "dictated" by subsets but an inherent property of the set itself.
This perspective aligns with rigorous mathematical reasoning: Infinity, though an abstract and unreachable concept in a constructive sense, remains well-defined and self-consistent within proper mathematical frameworks.
Best regards,
Soumendra Nath Thakur
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