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| 10482 | The logic of ZF is classical first-order predicate logic with identity |
| Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.121) | |||
| A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral. |
| 10492 | A few axioms of set theory 'force themselves on us', but most of them don't |
| Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.130) | |||
| A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture. |
| 10485 | Naïve sets are inconsistent: there is no set for things that do not belong to themselves |
| Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.127) | |||
| A reaction: As clear a summary of Russell's Paradox as you could ever hope for. |
| 10484 | The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first |
| Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.126) | |||
| A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent. |
| 10491 | Infinite natural numbers is as obvious as infinite sentences in English |
| Full Idea: The existence of infinitely many natural numbers seems to me no more troubling than that of infinitely many computer programs or sentences of English. There is, for example, no longest sentence, since any number of 'very's can be inserted. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |||
| A reaction: If you really resisted an infinity of natural numbers, presumably you would also resist an actual infinity of 'very's. The fact that it is unclear what could ever stop a process doesn't guarantee that the process is actually endless. |
| 10483 | Mathematics and science do not require very high orders of infinity |
| Full Idea: To the best of my knowledge nothing in mathematics or science requires the existence of very high orders of infinity. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.122) | |||
| A reaction: He is referring to particular high orders of infinity implied by set theory. Personally I want to wield Ockham's Razor. Is being implied by set theory a sufficient reason to accept such outrageous entities into our ontology? |
| 10490 | Mathematics isn't surprising, given that we experience many objects as abstract |
| Full Idea: It is no surprise that we should be able to reason mathematically about many of the things we experience, for they are already 'abstract'. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |||
| A reaction: He has just given a list of exemplary abstract objects (Idea 10489), but I think there is a more interesting idea here - that our experience of actual physical objects is to some extent abstract, as soon as it is conceptualised. |
| 10488 | It is lunacy to think we only see ink-marks, and not word-types |
| Full Idea: It's a kind of lunacy to think that sound scientific philosophy demands that we think that we see ink-tracks but not words, i.e. word-types. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |||
| A reaction: This seems to link him with Armstrong's mockery of 'ostrich nominalism'. There seems to be some ambiguity with the word 'see' in this disagreement. When we look at very ancient scratches on stones, why don't we always 'see' if it is words? |
| 10487 | I am a fan of abstract objects, and confident of their existence |
| Full Idea: I am rather a fan of abstract objects, and confident of their existence. Smaller numbers, sets and functions don't offend my sense of reality. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |||
| A reaction: The great Boolos is rather hard to disagree with, but I disagree. Logicians love abstract objects, indeed they would almost be out of a job without them. It seems to me they smuggle them into our ontology by redefining either 'object' or 'exists'. |
| 10489 | We deal with abstract objects all the time: software, poems, mistakes, triangles.. |
| Full Idea: We twentieth century city dwellers deal with abstract objects all the time, such as bank balances, radio programs, software, newspaper articles, poems, mistakes, triangles. | |||
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |||
| A reaction: I find this claim to be totally question-begging, and typical of a logician. The word 'object' gets horribly stretched in these discussions. We can create concepts which have all the logical properties of objects. Maybe they just 'subsist'? |