99 ideas
19608 | Wisdom is just the last gasp of a dying civilization [Cioran] |
19624 | Intelligence only fully flourishes at the end of a historical period [Cioran] |
19599 | Ideas are neutral, but people fill them with passion and weakness [Cioran] |
19631 | The history of ideas (and deeds) occurs in a meaningless environment [Cioran] |
19645 | Some thinkers would have been just as dynamic, no matter when they had lived [Cioran] |
19629 | A nation gives expression to its sum of values, and is then exhausted [Cioran] |
19618 | I abandoned philosophy because it didn't acknowledge melancholy and human weakness [Cioran] |
19621 | Originality in philosophy is just the invention of terms [Cioran] |
19607 | The mind is superficial, only concerned with the arrangement of events, not their significance [Cioran] |
19638 | Metaphysics is a universalisation of physical anguish [Cioran] |
19620 | Great systems of philosophy are just brilliant tautologies [Cioran] |
19630 | No great idea ever emerged from a dialogue [Cioran] |
19636 | Truth is just an error insufficiently experienced [Cioran] |
19642 | Eventually every 'truth' is guaranteed by the police [Cioran] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
13643 | Aristotelian logic is complete [Shapiro] |
13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
13644 | Semantics for models uses set-theory [Shapiro] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
13670 | Categoricity can't be reached in a first-order language [Shapiro] |
13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
19632 | An axiom has no more authority than a frenzy [Cioran] |
13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
13628 | We can live well without completeness in logic [Shapiro] |
13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
13646 | Compactness is derived from soundness and completeness [Shapiro] |
13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
19626 | Our instincts had to be blunted and diminished, to make way for consciousness! [Cioran] |
19633 | We use concepts to master our fears; saying 'death' releases us from confronting it [Cioran] |
19615 | I want to suppress in myself the normal reasons people have for action [Cioran] |
19628 | At a civilisation's peak values are all that matters, and people unconsciously live by them [Cioran] |
19646 | Values don't accumulate; they are ruthlessly replaced [Cioran] |
19614 | Lovers are hateful, apart from their hovering awareness of death [Cioran] |
19634 | Man is never himself; he always aims at less than life, or more than life [Cioran] |
19619 | To live authentically, we must see that philosophy is totally useless [Cioran] |
19622 | The pointlessness of our motives and irrelevance of our gestures reveals our vacuity [Cioran] |
19617 | Evidence suggests that humans do not have a purpose [Cioran] |
19612 | The universe is dirty and fragile, as if a scandal in nothingness had produced its matter [Cioran] |
19604 | Unlike other creatures, mankind seems lost in nature [Cioran] |
19606 | We can only live because our imagination and memory are poor [Cioran] |
19601 | Life is now more dreaded than death [Cioran] |
19640 | No one is brave enough to say they don't want to do anything; we despise such a view [Cioran] |
19602 | You are stuck in the past if you don't know boredom [Cioran] |
19641 | If you lack beliefs, boredom is your martyrdom [Cioran] |
19644 | History is the bloody rejection of boredom [Cioran] |
19613 | It is pointless to refuse or accept the social order; we must endure it like the weather [Cioran] |
19627 | Opportunists can save a nation, and heroes can ruin it [Cioran] |
19625 | The ideal is to impose a religion by force, and then live in doubt about its beliefs [Cioran] |
19605 | Despite endless suggestions, no one has found a goal for history [Cioran] |
19637 | History is wonderfully devoid of meaning [Cioran] |
19610 | Religions see suicide as insubordination [Cioran] |
19611 | No one has ever found a good argument against suicide [Cioran] |
19609 | If you have not contemplated suicide, you are a miserable worm [Cioran] |
19639 | We all need sexual secrets! [Cioran] |
19603 | Why is God so boring, and why does God resemble humanity so little? [Cioran] |
16713 | Philosophers are the forefathers of heretics [Tertullian] |
19616 | As the perfect wisdom of detachment, philosophy offers no rivals to Taoism [Cioran] |
19600 | When man abandons religion, he then follows new fake gods and mythologies [Cioran] |
19643 | A religion needs to motivate killings, and cannot tolerate rivals [Cioran] |
6610 | I believe because it is absurd [Tertullian] |
19623 | Circles of hell are ridiculous; all that matters is to be there [Cioran] |