21 ideas
| 23531 | Philosophers should interpret the world, by expressing its possibilities [Berardi] |
| Full Idea: The philosopher's task is to interpret the world, that is, to capture its tendency and above all to enunciate the possibilities inscribed therein. …The politician's eye does not see the possible, being attracted instead by the probable. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], How to) | |
| A reaction: An inspiring idea! He is rejecting Marx's aim of changing the world, which had 'catastrophic' results. But I love his view of interpretation as spotting tendencies and possibilities. This fits my preferred ontology of dispositions and powers. |
| 13030 | 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) |
| 13032 | 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. |
| 13033 | 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) |
| 13037 | 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) |
| 13038 | 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) |
| 13034 | 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) |
| 13039 | 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) |
| 13036 | 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) |
| 13029 | 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) |
| 13031 | 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. |
| 13040 | 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) |
| 23538 | Nothingness only exists in consciousness [Berardi] |
| Full Idea: Consciousness is the only place where nothingness exists. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], II 'Expanding') | |
| A reaction: Not sure about this, but an interesting remark from someone with a Hegelian background. We certainly have a concept of nothingness (a mental file of it, even). |
| 18465 | An 'equivalence' relation is one which is reflexive, symmetric and transitive [Kunen] |
| Full Idea: R is an equivalence relation on A iff R is reflexive, symmetric and transitive on A. | |
| From: Kenneth Kunen (The Foundations of Mathematics (2nd ed) [2012], I.7.1) |
| 23532 | The delusion of free will brings a sense of guilt [Berardi] |
| Full Idea: A sense of guilt is linked to the delusion of free will. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], How to) | |
| A reaction: I agree that free will is a delusion, but I'm not sure about this. Clearly if you think you are 'ultimately' responsible for all of your actions this will increase guilt, but belief in free will is compatible with various excuses for actions. |
| 4377 | Intellectualism is an excessive emphasis on reasoning in moral philosophy [Burnyeat] |
| Full Idea: Intellectualism, a one-sided preoccupation with reason and reasoning, is a perennial failing in moral philosophy. | |
| From: Myles F. Burnyeat (Aristotle on Learning to be Good [1980], p.70) | |
| A reaction: But Kant's reply would be that while there is much more to moral behaviour, the only part which matters in morality is the reasoning part. And Socrates' view (ignorance is evil) is not obviously wrong. |
| 23537 | American white men trusted the philosophy of winning, and then discovered losing [Berardi] |
| Full Idea: American white men trusted the promises of neoliberal selfishness, they trusted the philosophy of winning, then discovered losing. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], I 'Fascism') | |
| A reaction: The most pernicious terminology in the English-speaking world is the labelling of people as 'winners' and 'losers'. We celebrate the one winner, and ignore everyone else. Celebrity, excessive wealth, honourable titles. |
| 23536 | Community is now a nostalgic memory, which no longer exists [Berardi] |
| Full Idea: Community is only a nostalgic memory of a past condition of belonging that exists no longer. Regret, not a living experience. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], I 'Fascism') | |
| A reaction: It is hard to disagree with this. Local heroes in my town make lovely efforts to improve the place (with flowers, sculptures) but most of us don't know who they are. Capitalist competition erodes community. |
| 23534 | Communism failed to unite western workers with the oppressed of the south [Berardi] |
| Full Idea: Communist internationalism was the only attempt to reconcile the workers of the West and the oppressed population of the Global South, and this attempt failed. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], I 'Hundred') | |
| A reaction: It was unfortunate that communism was launched in Russia, which we now see (in 2023) as poisoned by imperialist ambitions, and quite unsuited to international idealism. The Chinese are notably active in Africa. |
| 23533 | The economy has replaced medieval theocracy at the centre of our society [Berardi] |
| Full Idea: The economy has progressively acquired the central place in the system of knowledge and research. Re-enacting the privilege of theocracy in the Middle Ages. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], I 'Knowledge') | |
| A reaction: Illuminating. This is indeed how the economy is treated, centring on Gross National Product (no matter how distributed), and economic league tables. Is it even a quasi-religion? |
| 23535 | Western workers turn to nationalism, to avert the effects of globalisation [Berardi] |
| Full Idea: Western workers are following nationalist agendas in order to avert the effects of globalization, and resorting to nationalist and racist forms of identification. | |
| From: Franco 'Bifo' Berardi (The Second Coming [2019], I 'Hundred') | |
| A reaction: By 'globalisation' must be meant the impersonal work and exploitation that results from huge ruthless anonymous companies. People barely know who they are working for, so it can't give them an identity. |