Combining Texts

All the ideas for 'works', 'Preface to Great Instauration (Renewal)' and 'Set Theory'

unexpand these ideas     |    start again     |     specify just one area for these texts


15 ideas

1. Philosophy / D. Nature of Philosophy / 7. Despair over Philosophy
Philosophy is like a statue which is worshipped but never advances [Bacon]
     Full Idea: Philosophy and the intellectual sciences stand like statues, worshipped and celebrated, but not moved or advanced.
     From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], Vol.4.14), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5
     A reaction: Still the view of most scientists, I suspect. Personally I disagree, because I think philosophy has made enormous advances, in accurate analysis of arguments. The trouble is there is so much of it that it is hard to discern, and we don't live long enough.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen]
     Full Idea: Axiom of Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y). That is, a set is determined by its members. If every z in one set is also in the other set, then the two sets are the same.
     From: Kenneth Kunen (Set Theory [1980], §1.5)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen]
     Full Idea: Axiom of Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z). Any pair of entities must form a set.
     From: Kenneth Kunen (Set Theory [1980], §1.6)
     A reaction: Repeated applications of this can build the hierarchy of sets.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen]
     Full Idea: Axiom of Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A). That is, the union of a set (all the members of the members of the set) must also be a set.
     From: Kenneth Kunen (Set Theory [1980], §1.6)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen]
     Full Idea: Axiom of Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x). That is, there is a set which contains zero and all of its successors, hence all the natural numbers. The principal of induction rests on this axiom.
     From: Kenneth Kunen (Set Theory [1980], §1.7)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen]
     Full Idea: Power Set Axiom: ∀x ∃y ∀z(z ⊂ x → z ∈ y). That is, there is a set y which contains all of the subsets of a given set. Hence we define P(x) = {z : z ⊂ x}.
     From: Kenneth Kunen (Set Theory [1980], §1.10)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen]
     Full Idea: Axiom of Replacement Scheme: ∀x ∈ A ∃!y φ(x,y) → ∃Y ∀X ∈ A ∃y ∈ Y φ(x,y). That is, any function from a set A will produce another set Y.
     From: Kenneth Kunen (Set Theory [1980], §1.6)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen]
     Full Idea: Axiom of Foundation: ∀x (∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))). Aka the 'Axiom of Regularity'. Combined with Choice, it means there are no downward infinite chains.
     From: Kenneth Kunen (Set Theory [1980], §3.4)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Choice: ∀A ∃R (R well-orders A) [Kunen]
     Full Idea: Axiom of Choice: ∀A ∃R (R well-orders A). That is, for every set, there must exist another set which imposes a well-ordering on it. There are many equivalent versions. It is not needed in elementary parts of set theory.
     From: Kenneth Kunen (Set Theory [1980], §1.6)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
Set Existence: ∃x (x = x) [Kunen]
     Full Idea: Axiom of Set Existence: ∃x (x = x). This says our universe is non-void. Under most developments of formal logic, this is derivable from the logical axioms and thus redundant, but we do so for emphasis.
     From: Kenneth Kunen (Set Theory [1980], §1.5)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen]
     Full Idea: Comprehension Scheme: for each formula φ without y free, the universal closure of this is an axiom: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ). That is, there must be a set y if it can be defined by the formula φ.
     From: Kenneth Kunen (Set Theory [1980], §1.5)
     A reaction: Unrestricted comprehension leads to Russell's paradox, so restricting it in some way (e.g. by the Axiom of Specification) is essential.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
Constructibility: V = L (all sets are constructible) [Kunen]
     Full Idea: Axiom of Constructability: this is the statement V = L (i.e. ∀x ∃α(x ∈ L(α)). That is, the universe of well-founded von Neumann sets is the same as the universe of sets which are actually constructible. A possible axiom.
     From: Kenneth Kunen (Set Theory [1980], §6.3)
12. Knowledge Sources / B. Perception / 1. Perception
The senses deceive, but also show their own errors [Bacon]
     Full Idea: It is certain that the senses deceive, but they also testify to their own errors.
     From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], p.32), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.1
     A reaction: Nice. This is the empiricist view, rather than the rationalist line that reason sorts out the mess created by the senses. Most people know things if you just show them.
14. Science / A. Basis of Science / 3. Experiment
Nature is revealed when we put it under pressure rather than observe it [Bacon]
     Full Idea: The secrets of nature reveal themselves more readily under the vexations of art than when they go their own way.
     From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], Vol.4.95), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5
     A reaction: This is a splendid slogan for the dawn of the age of science, and pinpoints the reason why we have advanced so much further than the Greeks. You can, of course, overdo the 'vexations of art'. It also justifies the critical approach to philosophy.
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
There are two sides to men - the pleasantly social, and the violent and creative [Diderot, by Berlin]
     Full Idea: Diderot is among the first to preach that there are two men: the artificial man, who belongs in society and seeks to please, and the violent, bold, criminal instinct of a man who wishes to break out (and, if controlled, is responsible for works of genius.
     From: report of Denis Diderot (works [1769], Ch.3) by Isaiah Berlin - The Roots of Romanticism
     A reaction: This has an obvious ancestor in Plato's picture (esp. in 'Phaedrus') of the two conflicting sides to the psuché, which seem to be reason and emotion. In Diderot, though, the suppressed man has virtues, which Plato would deny.