54 ideas
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
3942 | I do not believe in the existence of anything, if I see no reason to believe it [Berkeley] |
3952 | I know that nothing inconsistent can exist [Berkeley] |
3959 | There is no other substance, in a strict sense, than spirit [Berkeley] |
12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
3946 | A thing is shown to be impossible if a contradiction is demonstrated within its definition [Berkeley] |
3958 | Since our ideas vary when the real things are said to be unchanged, they cannot be true copies [Berkeley] |
3943 | If existence is perceived directly, by which sense; if indirectly, how is it inferred from direct perception? [Berkeley] |
3931 | Sensible objects are just sets of sensible qualities [Berkeley] |
5192 | Berkeley did not deny material things; he merely said they must be defined through sensations [Berkeley, by Ayer] |
5174 | Berkeley needed a phenomenalist account of the self, as well as of material things [Ayer on Berkeley] |
1103 | 'To be is to be perceived' is a simple confusion of experience with its objects [Russell on Berkeley] |
6403 | For Berkelely, reality is ideas and a community of minds, including God's [Berkeley, by Grayling] |
3936 | Time is measured by the succession of ideas in our minds [Berkeley] |
3930 | There is no such thing as 'material substance' [Berkeley] |
3939 | I conceive a tree in my mind, but I cannot prove that its existence can be conceived outside a mind [Berkeley] |
3945 | There is nothing in nature which needs the concept of matter to explain it [Berkeley] |
3947 | Perceptions are ideas, and ideas exist in the mind, so objects only exist in the mind [Berkeley] |
3933 | Primary qualities (such as shape, solidity, mass) are held to really exist, unlike secondary qualities [Berkeley] |
3934 | A mite would see its own foot as large, though we would see it as tiny [Berkeley] |
3935 | The apparent size of an object varies with its distance away, so that can't be a property of the object [Berkeley] |
3937 | 'Solidity' is either not a sensible quality at all, or it is clearly relative to our senses [Berkeley] |
3940 | Distance is not directly perceived by sight [Berkeley] |
3957 | Immediate objects of perception, which some treat as appearances, I treat as the real things themselves [Berkeley] |
3953 | Real things and imaginary or dreamed things differ because the latter are much fainter [Berkeley] |
3938 | Geometry is originally perceived by senses, and so is not purely intellectual [Berkeley] |
3944 | It is possible that we could perceive everything as we do now, but nothing actually existed. [Berkeley] |
3932 | A hot hand and a cold hand will have different experiences in the same tepid water [Berkeley] |
3948 | Experience tells me that other minds exist independently from my own [Berkeley] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
3941 | How can that which is unthinking be a cause of thought? [Berkeley] |
5374 | Berkeley probably used 'idea' to mean both the act of apprehension and the thing apprehended [Russell on Berkeley] |
3954 | Immorality is not in the action, but in the deviation of the will from moral law [Berkeley] |
3950 | There must be a God, because all sensible things must be perceived by him [Berkeley] |
3951 | There must be a God, because I and my ideas are not independent [Berkeley] |
3949 | It has been proved that creation is the workmanship of God, from its beauty and usefulness [Berkeley] |
3956 | People are responsible because they have limited power, though this ultimately derives from God [Berkeley] |
3955 | If sin is not just physical, we don't consider God the origin of sin because he causes physical events [Berkeley] |