6 ideas
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 13342 | Carnap defined consequence by contradiction, but this is unintuitive and changes with substitution [Tarski on Carnap] |
| Full Idea: Carnap proposed to define consequence as 'sentence X follows from the sentences K iff the sentences K and the negation of X are contradictory', but 1) this is intuitively impossible, and 2) consequence would be changed by substituting objects. | |
| From: comment on Rudolph Carnap (The Logical Syntax of Language [1934], p.88-) by Alfred Tarski - The Concept of Logical Consequence p.414 | |
| A reaction: This seems to be the first step in the ongoing explicit discussion of the nature of logical consequence, which is now seen by many as the central concept of logic. Tarski brings his new tool of 'satisfaction' to bear. |
| 13251 | Each person is free to build their own logic, just by specifying a syntax [Carnap] |
| Full Idea: In logic, there are no morals. Everyone is at liberty to build his own logic, i.e. his own form of language. All that is required is that he must state his methods clearly, and give syntactical rules instead of philosophical arguments. | |
| From: Rudolph Carnap (The Logical Syntax of Language [1934], §17), quoted by JC Beall / G Restall - Logical Pluralism 7.3 | |
| A reaction: This is understandable, but strikes me as close to daft relativism. If I specify a silly logic, I presume its silliness will be obvious. By what criteria? I say the world dictates the true logic, but this is a minority view. |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |