Combining Texts

All the ideas for 'Set Theory', 'Could a computer ever understand?' and 'True in Theory, but not in Practice'

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26 ideas

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]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / e. Cause of consciousness
4921 Quantum states in microtubules could bind brain activity to produce consciousness [Penrose]
20. Action / C. Motives for Action / 3. Acting on Reason / a. Practical reason
21059 General rules of action also need a judgement about when to apply them [Kant]
22. Metaethics / C. The Good / 2. Happiness / c. Value of happiness
21061 Duty does not aim at an end, but gives rise to universal happiness as aim of the will [Kant]
23. Ethics / D. Deontological Ethics / 2. Duty
21060 It can't be a duty to strive after the impossible [Kant]
23. Ethics / D. Deontological Ethics / 6. Motivation for Duty
21062 The will's motive is the absolute law itself, and moral feeling is receptivity to law [Kant]
24. Political Theory / A. Basis of a State / 3. Natural Values / c. Natural rights
21071 There can be no restraints on freedom if reason does not reveal some basic rights [Kant]
24. Political Theory / B. Nature of a State / 2. State Legitimacy / c. Social contract
21063 Personal contracts are for some end, but a civil state contract involves a duty to share [Kant]
21068 There must be a unanimous contract that citizens accept majority decisions [Kant]
21069 A contract is theoretical, but it can guide rulers to make laws which the whole people will accept [Kant]
24. Political Theory / B. Nature of a State / 2. State Legitimacy / d. General will
21070 A law is unjust if the whole people could not possibly agree to it [Kant]
24. Political Theory / B. Nature of a State / 4. Citizenship
21067 A citizen must control his own life, and possess property or an important skill [Kant]
24. Political Theory / D. Ideologies / 6. Liberalism / a. Liberalism basics
21064 A lawful civil state must embody freedom, equality and independence for its members [Kant]
25. Social Practice / B. Equalities / 4. Economic equality
21066 Citizens can rise to any rank that talent, effort and luck can achieve [Kant]
25. Social Practice / C. Rights / 3. Alienating rights
21065 You can't make a contract renouncing your right to make contracts! [Kant]
25. Social Practice / E. Policies / 1. War / a. Just wars
21072 The people (who have to fight) and not the head of state should declare a war [Kant]