56 ideas
| 15053 | If metaphysics can't be settled, it hardly matters whether it makes sense [Fine,K] |
| 15054 | 'Quietist' says abandon metaphysics because answers are unattainable (as in Kant's noumenon) [Fine,K] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 15007 | If you make 'grounding' fundamental, you have to mention some non-fundamental notions [Sider on Fine,K] |
| 15006 | Something is grounded when it holds, and is explained, and necessitated by something else [Fine,K, by Sider] |
| 15055 | Grounding relations are best expressed as relations between sentences [Fine,K] |
| 15050 | Reduction might be producing a sentence which gets closer to the logical form [Fine,K] |
| 15051 | Reduction might be semantic, where a reduced sentence is understood through its reduction [Fine,K] |
| 15052 | Reduction is modal, if the reductions necessarily entail the truth of the target sentence [Fine,K] |
| 15056 | The notion of reduction (unlike that of 'ground') implies the unreality of what is reduced [Fine,K] |
| 15060 | Why should what is explanatorily basic be therefore more real? [Fine,K] |
| 15046 | Reality is a primitive metaphysical concept, which cannot be understood in other terms [Fine,K] |
| 15047 | What is real can only be settled in terms of 'ground' [Fine,K] |
| 15048 | In metaphysics, reality is regarded as either 'factual', or as 'fundamental' [Fine,K] |
| 10938 | The extremes of essentialism are that all properties are essential, or only very trivial ones [Rami] |
| 10940 | An 'individual essence' is possessed uniquely by a particular object [Rami] |
| 10939 | 'Sortal essentialism' says being a particular kind is what is essential [Rami] |
| 10934 | Unlosable properties are not the same as essential properties [Rami] |
| 10933 | Physical possibility is part of metaphysical possibility which is part of logical possibility [Rami] |
| 10932 | If it is possible 'for all I know' then it is 'epistemically possible' [Rami] |
| 15061 | Although colour depends on us, we can describe the world that way if it picks out fundamentals [Fine,K] |
| 15059 | Grounding is an explanation of truth, and needs all the virtues of good explanations [Fine,K] |
| 15057 | Ultimate explanations are in 'grounds', which account for other truths, which hold in virtue of the grounding [Fine,K] |
| 15058 | A proposition ingredient is 'essential' if changing it would change the truth-value [Fine,K] |