Zeno's Paradoxes and the Concept of Infinity

Dichotomy Paradox

Achilles never reaches the finish line due to infinite divisions.

Infinite Segments

Total length is a sum of infinitely divided parts.

Limits in Calculus

Method for defining infinite sums and convergence.

Finite Arithmetic

Does not apply to infinite series calculations.

Cauchy's Contributions

Formalized limits, derivatives, and continuity in calculus.

Infinitesimals

Numbers smaller than any finite value, not zero.

Berkeley's Criticism

Infinitesimals termed 'ghosts of departed quantities'.

Hilbert's Paradox

Infinite places can accommodate infinite guests.

Cantor's Diagonal Argument

Proves real numbers are uncountably infinite.

Cantor's Theorem

Power set of any set is larger than the set.

Mereology

Study of part-whole relationships in objects.

Ship of Theseus Paradox

Questions identity with gradual part replacement.

Leibniz's Law

Identical properties imply identity between objects.

Identity Over Time

Concept of persistence and change in objects.

Material Objects

Entities that occupy space, persist, and have mass.

Two-Object View

Lump and statue coexist as distinct entities.

Zeno's Reasoning Flaw

Diogenes demonstrated motion despite Zeno's argument.

Mathematical Solution

Calculus resolves paradoxes through rigorous definitions.

Natural Numbers and Integers

Same cardinality via one-to-one correspondence.

Measure Theory

Defines total length from infinite small values.

Paradox of Plurality

Segments composed of infinite points challenge length.