Combining Texts

All the ideas for 'Set Theory', 'The Thesis that Mathematics is Logic' and 'The Art of Rhetoric'

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29 ideas

2. Reason / A. Nature of Reason / 1. On Reason
23250 Desired responsible actions result either from rational or from irrational desire [Aristotle]
2. Reason / C. Styles of Reason / 1. Dialectic
5847 It is the role of dialectic to survey syllogisms [Aristotle]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
13030 Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
13032 Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
13033 Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
13037 Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
13038 Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
13034 Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
13039 Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13036 Choice: ∀A ∃R (R well-orders A) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
13029 Set Existence: ∃x (x = x) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
13031 Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
13040 Constructibility: V = L (all sets are constructible) [Kunen]
5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
10066 Putnam coined the term 'if-thenism' [Putnam, by Musgrave]
14. Science / A. Basis of Science / 6. Falsification
5862 A single counterexample is enough to prove that a truth is not necessary [Aristotle]
14. Science / C. Induction / 1. Induction
5854 Nobody fears a disease which nobody has yet caught [Aristotle]
19. Language / F. Communication / 1. Rhetoric
5849 Rhetoric is a political offshoot of dialectic and ethics [Aristotle]
21. Aesthetics / A. Aesthetic Experience / 5. Natural Beauty
5851 Pentathletes look the most beautiful, because they combine speed and strength [Aristotle]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
5858 Men are physically prime at thirty-five, and mentally prime at forty-nine [Aristotle]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention
5855 We all feel universal right and wrong, independent of any community or contracts [Aristotle]
22. Metaethics / C. The Good / 2. Happiness / d. Routes to happiness
5850 Happiness is composed of a catalogue of internal and external benefits [Aristotle]
23. Ethics / A. Egoism / 1. Ethical Egoism
5856 Self-interest is a relative good, but nobility an absolute good [Aristotle]
23. Ethics / C. Virtue Theory / 1. Virtue Theory / a. Nature of virtue
5853 The best virtues are the most useful to others [Aristotle]
5848 All good things can be misused, except virtue [Aristotle]
23. Ethics / C. Virtue Theory / 3. Virtues / f. Compassion
5857 The young feel pity from philanthropy, but the old from self-concern [Aristotle]
23. Ethics / C. Virtue Theory / 4. External Goods / c. Wealth
5859 Rich people are mindlessly happy [Aristotle]
24. Political Theory / B. Nature of a State / 3. Constitutions
5852 The four constitutions are democracy (freedom), oligarchy (wealth), aristocracy (custom), tyranny (security) [Aristotle]
25. Social Practice / D. Justice / 3. Punishment / b. Retribution for crime
1660 It is noble to avenge oneself on one's enemies, and not come to terms with them [Aristotle]
26. Natural Theory / C. Causation / 5. Direction of causation
5861 People assume events cause what follows them [Aristotle]