21 ideas
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 8228 | Being is revealed at the point between waking and sleep [Anon (Cent)] |
| Full Idea: At the point of sleep when sleep has not yet come and external wakefulness vanishes, at this point being is revealed. | |
| From: Anon (Cent) (Centring [c.500 BCE], 50) | |
| A reaction: One for Heidegger. The problem with eastern philosophy is that no one expects that this revelation could be put into words, so we cannot compare our revelations of being, or pass them on to others. Which means they can't be criticised. |
| 8232 | Neglect your self, and feel the consciousness of each other being [Anon (Cent)] |
| Full Idea: Feel the consciousness of each person as your own consciousness. So, leaving aside concern for self, become each being. | |
| From: Anon (Cent) (Centring [c.500 BCE], 83) | |
| A reaction: Good. The western scepticism about other minds puts up a barrier to this sort of thought. The best of Zen is when it encourages the use of imagination to explore what is real. But imagination struggles to distinguish what is true. |
| 8230 | Just as you have the impulse to do something, stop [Anon (Cent)] |
| Full Idea: Just as you have the impulse to do something, stop. | |
| From: Anon (Cent) (Centring [c.500 BCE], 64) | |
| A reaction: Nice. You might train your own will like a dog in this way. It is a counterexample to the simplistic idea that all you need is a belief and a desire, and you have got an action. But (pace Searle, Ideas 3817 + 3818) this does not prove free will. |
| 5996 | Critolaus redefined Aristotle's moral aim as fulfilment instead of happiness [Critolaus, by White,SA] |
| Full Idea: Critolaus reformulated Aristotelian theory by defining happiness as a 'fulfilment' (sumplêrôma) of psychic, physical, and external goods, where virtue vastly outweighs the rest. | |
| From: report of Critolaus (fragments/reports [c.170 BCE]) by Stephen A. White - Critolaus | |
| A reaction: The sounds more like an attempt at clarification than a real change of Peripatetic doctrine. Occasionally 'fulfilment' is offered as a translation for eudaimonia. Maybe we should just take up Critolaus' suggestion when we are discussing Aristotle. |
| 8234 | Bondage and liberation are relative terms, which only frighten those already terrified of the universe [Anon (Cent)] |
| Full Idea: Since in truth, bondage and freedom are relative, these words are only for those terrified with the universe. The universe is a reflection of minds. As you see many suns in water from one sun, so see bondage and liberation. | |
| From: Anon (Cent) (Centring [c.500 BCE], 110) | |
| A reaction: Since most easterners have experienced permanent social repression, their philosophies all encourage them not to worry about it. No wonder marxism, proved popular, when it suggested you could actually change things. |
| 8227 | Feel your whole body saturated with cosmic essence [Anon (Cent)] |
| Full Idea: Feel your substance, bones, flesh, blood, saturated with cosmic essence. | |
| From: Anon (Cent) (Centring [c.500 BCE], 23) | |
| A reaction: I find this unsatisfactory. Being a sad victim of materialistic western scientific culture, I don't think 'cosmic essence' exists. I could imagine myself full of the stuff, and that might feel good, but I could also imagine I had won the lottery. |
| 8231 | Abandon attachment to body, and feel the joy of being everywhere [Anon (Cent)] |
| Full Idea: Toss attachment for body aside, realising I am everywhere. One who is everywhere is joyous. | |
| From: Anon (Cent) (Centring [c.500 BCE], 79) | |
| A reaction: I really don't think it is wisdom to spend your time imagining things which are not true. I might imagine that some gorgeous film star is in love with me, but I don't recommend it. Live according to nature. We are physical beings. |
| 8229 | The serenity in blue sky beyond clouds [Anon (Cent)] |
| Full Idea: Simply by looking into blue sky beyond clouds - the serenity. | |
| From: Anon (Cent) (Centring [c.500 BCE], 59) | |
| A reaction: Philip Larkin finds the same experience looking through high windows. If we articulate the experience, it seems to combine humility and detachment. The sky makes us insignificant, but the mind can project into the sky. |
| 8233 | Imagine fire burning up your whole body, but not you [Anon (Cent)] |
| Full Idea: Focus on fire rising through your form from the toes up until the body burns to ashes, but not you. | |
| From: Anon (Cent) (Centring [c.500 BCE], 28) | |
| A reaction: Try very hard to persuade yourself that you are immortal, even if you are not. Hm. Zen Buddhism actually contains very little commitment to what is true or false. It explores interesting beliefs like children playing with toys. Very post-modern. |